Cette thèse porte sur les statistiques bayésiennes non paramétriques, avec une attention particulière pour les modèles de mélanges non paramétriques. La thèse est divisée en une introduction générale et trois parties traitant des aspects relativement différents des approches par mélanges (échantillonage, asymptotique, problème inverse). Un résumé des différents chapitres est proposé ci-dessous.

Chapitre 1 (Introduction) Dans un premier temps, nous présentons les concepts clés étudiés dans cette thèse. Particulièrement, nous discutons de l'analyse fréquentiste des méthodes bayésiennes, un des thèmes récurrent au cours des chapitres qui suivent. Dans un second temps, nous nous intéressons à une classe particulière de modèles, à savoir les modèles de mélange non paramétriques, thème central de la thèse. Les bases des modèles de mélange sont données, avec un intérêt particulier pour les mélanges par processus Gamma symétriques. La troisème partie de ce chapitre est dédiée à l'échantillonage, un concept clé dans le chapitre 2. Finalement, nous introduisons les connaissances nécessaires à la bonne compréhension du chapitre 4, où les outils développés au long de cette thèse sont utilisés pour résoudre le problème inverse mal posé de la Tomographie Quantique Homodyne.

Chapitre 2 (co-écrit avec Éric Barat) Dans cet article, nous présentons quelques aspects spécifiques aux mélanges par processus Gamma symétriques pour un usage en régression. Nous proposons un nouvel échantilloneur de Gibbs pour simuler la distribution a posteriori et nous établissons des vitesses de convergence adaptatives pour le modèle de régression moyenne avec bruit Gaussien.

Chapitre 3 (co-écrit avec Judith Rousseau) De nos jours en estimation de densité, les vitesses de contraction de la distribution a posteriori pour les mélanges en translation et en moyenne et variance de Gaussiennes ne sont connus que sous des hypothèses de queues légères; avec de meilleures vitesses obtenues pour les mélanges en translation. Il a été conjecturé par Canale et DeBlasi, mais non prouvé, que la situation devrait être inversée sous des hypothèses de queues lourdes. Cette conjecture est basée sur l'intuition qu'il n'est pas nécessaire d'obtenir un bon ordre d'approximation dans les régions avec peu d'observations (disons les queues), favorisant l'usage des mélanges en moyenne et variance qui peuvent adapter l'ordre d'approximation spatialement. Ici, nous testons l'argument précédent sur le problème de régression gaussienne avec covariables aléatoires. Bien que nous ne puissions pas encore invalider la conjecture, nous trouvons que même avec une hypothèse de queues lourdes, les mélanges en moyenne et variance semblent être moins performants que les mélanges en translation. Cependant, les preuves suggèrent 1 l'introduction d'un mélange hybride, qui semble être toujours plus performant que les deux autres, qu'importe la nature des queues. Finalement, nous montrons que toutes les hypothèses de queues peuvent être supprimées, au prix de rendre la distribution a priori dépendente des covariables.

Chapitre 4 (co-écrit avec Éric Barat) Nous estimons l'état quantique d'un faisceau de lumière à partir des résultats bruités de mesures de tomographie quantique homodyne, réalisées sur des systèmes quantiques identiquement préparés. Nous proposons deux approches Bayésiennes non paramétriques. La première approche est basée sur les modèles de mélanges et est illustrée au travers d'exemples de simulation. La seconde approche est basée sur une expansion sur une base. Nous étudions les performances théoriques de la seconde approche en quantifiant la vitesse de contraction de la distribution a posteriori autour du vrai état quantique dans la métrique L 2 .

Summary

This dissertation deals with Bayesian nonparametric statistics, in particular nonparametric mixture models. The manuscript is divided into a general introduction and three parts on rather different aspects of mixtures approaches (sampling, asymptotic, inverse problem). A summary of each chapter is proposed below.

Chapter 1 (Introduction) In a first time, we present the core concepts studied throughout this thesis. Particularly, we discuss frequentist analysis of Bayesian procedures, which constitute one of the governing principle of the next chapters. In a second time, we focus on a particular class of models, namely nonparametric mixture models, which is central to the thesis. Basics of mixture models are given, with a stress on Symmetric Gamma Processes Mixtures. The third section of this chapter is dedicated to posterior sampling, a key concept in chapter 2. Finally, we introduce the material required for the proper understanding of chapter 4, where the tools developed along the thesis are used to solve the ill-posed inverse problem of Quantum Homodyne Tomography.

Chapter 2 (joint work with Éric Barat) In this article, we present some specific aspects of symmetric Gamma process mixtures for use in regression models. We propose a new Gibbs sampler for simulating the posterior and we establish adaptive posterior rates of convergence related to the Gaussian mean regression problem.

Chapter 3 (joint work with Judith Rousseau) Nowadays in density estimation, posterior rates of convergence for location and location-scale mixtures of Gaussians are only known under light-tail assumptions; with better rates achieved by location mixtures. It has been conjectured by Canale and DeBlasi, but not proved, that the situation should be reversed under heavy tails assumptions. The conjecture is based on the feeling that there is no need to achieve a good order of approximation in regions with few data (say, in the tails), favoring location-scale mixtures which allow for spatially varying order of approximation. Here we test the previous argument on the Gaussian errors mean regression model with random design, for which the light tail assumption is not required for proofs. Although we cannot invalidate the conjecture due to the lack of lower bound, we find that even with heavy tails assumptions, location-scale mixtures apparently perform in general worst than location mixtures. However, the proofs suggest to introduce hybrid location-scale mixtures that are find to outperform both location and location-scale mixtures, whatever the nature of the tails. Finally, we show that all tails assumptions can be released at the price of making the prior distribution covariate dependent.

Je tiens à remercier Judith Rousseau pour m'avoir encadré et transmis une partie de ses connaissances pendant ces trois années et demi de thèse. Merci pour ta patience, en particulier au début de ma thèse, et merci pour toutes les fois où ton intuition géniale m'a évité de perdre des semaines entières de travail. Avec le recul, je mesure la chance que j'ai eu de pouvoir travailler à tes côtés.

J'exprime également toute ma gratitude à Truong Trong-Tuong pour avoir accepté de co-encadrer ce travail, et pour sa bienveillance et son soutient tout au long de cette thèse.

L'essentiel du travail présenté dans ce manuscrit a été réalisé au laboratoire LM2S du Commissariat à l'Energie Atomique (CEA), où j'ai passé trois ans. Un immense merci à Éric Barat pour avoir initié ce sujet de thèse et pour m'avoir choisi comme doctorant. Je te suis infiniment reconnaissant pour la liberté dont j'ai joui durant mes années au CEA. Merci pour avoir eu confiance en moi depuis le début et de m'avoir permis de travailler sur un sujet aussi extraordinaire. Il est clair que sans toi, cette thèse ne serait pas ce qu'elle est aujourd'hui.

Je remercie également toute l'équipe du LM2S, ainsi que tous ceux qui sont passés au laboratoire pendant la période de ma thèse. Merci pour tous les échanges constructifs, les moments partagés autour d'un café, un repas, un billard... J'adresse mes remerciements à Ismael Castillo et Surya Tokdar pour avoir accepté de rapporter cette thèse. Je suis très honoré de l'intérêt et de l'enthousiasme que vous avez manifesté pour ce travail. Je vous suis reconnaissant pour le temps passé, ainsi que pour vos remarques et suggestions. Merci également à Cristina Butucea, Dominique Picard et Vincent Rivoirard d'avoir accepté de compléter ce jury.

Durant ces trois années, j'ai fait partie de l'ANR BNPSI, une expérience riche qui m'a permis de rencontrer des personnes formidables avec qui nous avons pu échanger sur nos travaux au cours de réunions mémorables. Un grand merci à eux! En juin 2015, j'ai eu la chance de visiter le département de statistiques de l'Université de Duke. Je tenais à remercier Robert Wolpert et Merlyse Clyde pour leur invitation. Je remercie également tout le département pour leur accueil remarquable.

Une partie de ce manuscrit a été rédigée lors de mon séjour à l'Université d'Oxford en mai-juin dernier. J'en profite pour remercier tous les membres du laboratoire, en particulier François Caron qui est à l'origine de mon séjour là-bas.

Je voudrai également remercier toutes les personnes ayant soutenu et apprécié mon travail, et plus particulièrement ma famille qui m'a toujours soutenu, encouragé et aidé.

Mes derniers remerciements vont à Jessica pour son soutien inconditionnel. Ton support m'a été et me sera toujours très cher. Cette thèse et moi te devons beaucoup.

Asymptotic comparison

Special functions

Introduction

Résumé

Dans un premier temps, nous présentons les concepts clés étudiés dans cette thèse. Particulièrement, nous discutons de l'analyse fréquentiste des méthodes bayésiennes, un des thèmes récurrent au cours des chapitres qui suivent. Dans un second temps, nous nous intéressons à une classe particulière de modèles, à savoir les modèles de mélange non paramétriques, thème central de la thèse.

Les bases des modèles de mélange sont données, avec un intérêt particulier pour les mélanges par processus Gamma symétriques. La troisème partie de ce chapitre est dédiée à l'échantillonage, un concept clé dans le chapitre 2. Finalement, nous introduisons les connaissances nécessaires à la bonne compréhension du chapitre 4, où les outils développés au long de cette thèse sont utilisés pour résoudre le problème inverse mal posé de la Tomographie Quantique Homodyne.

Chapter 4 (joint work with Éric Barat) We estimate the quantum state of a light beam from results of quantum homodyne tomography noisy measurements performed on identically prepared quantum systems. We propose two Bayesian nonparametric approaches. The first approach is based on mixture models and is illustrated through simulation examples. The second approach is based on random basis expansions. We study the theoretical performance of the second approach by quantifying the rate of contraction of the posterior distribution around the true quantum state in the L 2 metric. The minimum, respectively the maximum, of a and b (x) + 0 if x ≤ 0, x otherwise X × Y Cartesian product of sets X and Y

List of symbols

Measure theory

(Ω, A) Measurable space (Ω, A, P)

Measured space, probability space if P is a probability measure B X , B(X)

Borel σ-algebra of the topological space X

E

Expectation operator P f

Abbreviate notation for f dP

Σ 1 ⊗ Σ 2 Product σ-algebra |µ|
Total variation measure of the signed measure µ X ∼ µ

The random variable X has distribution µ

Function spaces

S(X)

The Schwartz space over X L p (X, X , µ)

The L p space over X, X , µ C β (X)

The β-Hölder space over X • p

The L p norm Whenever convenient, we use the abbreviations S, L p , L p (X), and C β . The existence of regular conditional probability measures for Π n is not guaranteed without further assumptions of X n × Θ. In the sequel, we shall always assume that both X n and Θ are complete separable metric spaces, which ensure the existence of regular conditional distributions for Π n (Dudley, 2002, theorem 10.2.2).

Linear algebra

Given a model (X n ×Θ, B n ⊗Σ θ , Π n ), we define

P (n) θ (•) := Π n (• | θ)
as the conditional distribution of the observations X n ∈ X n given the parameter θ ∈ Θ. The marginal distribution Π n,θ (•) on Θ of Π n is called the prior distribution, and reflects the prior belief in θ in absence of any observation. The marginal distribution P Π of X n is called the prior predictive distribution.

The goal of the Bayesian inference is to characterize the posterior distribution, defined as Π n,θ (• | X n ), the conditional distribution of θ given X n ∈ X n . A common framework is when the family {P (n) θ : θ ∈ Θ} is dominated by a measure λ n on (X n , B n ). Then a regular version of the posterior distribution is given by the Bayes rule. : θ ∈ Θ} is dominated by a measure λ n . Then a regular version of the posterior distribution Π n,θ (• | X n ) is given by the following expression. For all

A ∈ Σ θ Π n,θ (θ ∈ A | X n ) = A dP (n) θ dλ n (X n ) dΠ n,θ (θ) Θ dP (n) θ dλ n (X n ) dΠ n,θ (θ).
The transition from prior distribution to posterior distribution represents the way in which prior beliefs are turned onto posterior beliefs based on the data. Theorem 1.1 and definition 1.1 emphasize the fact that in the Bayesian paradigm, all values of θ in the support of Π n,θ are possible with more or less likeliness, even in absence of data.

Finally, let mention that although the prior distribution Π n,θ is allowed to depend on n, this will not be the case in this dissertation, except at the end of chapter 3. When the prior distributions considered have some dependency on n, it will be mentioned explicitly; otherwise it is assumed that priors have no dependency on n.

For further discussion on foundations of the Bayesian approach, we refer to [START_REF] Robert | The Bayesian choice: from decision-theoretic foundations to computational implementation[END_REF]; [START_REF] Bernardo | Model selection via testing: an alternative to (penalized) maximum likelihood estimators[END_REF]; [START_REF] Berger | Statistical decision theory and Bayesian analysis[END_REF]; Van der Vaart (2000); Le [START_REF] Cam | Asymptotics in statistics: some basic concepts[END_REF]; [START_REF] Cam | Asymptotic methods in statistical decision theory[END_REF].

Frequentist validation of Bayes procedures

One of the most frequent criticisms about the Bayesian approach concerns the arbitrariness in the choice of the prior distribution. Of course, when there is concrete prior knowledge about the parameter, the Bayesian paradigm provide a natural way to incorporate it into the model, and this must be done. However, in many cases, prior knowledge is vague, or incomplete, which makes very difficult to specify a well justified prior distribution. Different people, with different opinions, may specify different prior distributions and arrive at different conclusions.

Frequentist study of the posterior distribution is a way to address the above concern. Indeed, the goal is to find conditions on the model under which, with a sufficient amount of data, the sequence of posterior distributions concentrates (in a sense to be clarified below) around the true θ 0 in the frequentist sense.

Introduction

We should note that the argument in the previous paragraph is not the only reason why a Bayesian should take care about frequentist properties of the posterior distribution. See [START_REF] Diaconis | On the consistency of bayes estimates[END_REF] for a discussion. Now we clarify the frequentist properties that we consider here, from the weaker to the stronger notion.

Definition 1.2 -Posterior consistency. -Let Θ be endowed with the topology T Θ . The sequence of posterior distributions {Π n,θ (• | X n ) : n ≥ 1} is said to be consistent for T Θ at θ 0 if for any neighborhood U of θ 0 we have P (n)

θ 0 Π n,θ (U | X n ) → 1.
Definition 1.3 -Rates of convergence. -Let Θ be endowed with the semimetric d n , and let ( n ) n≥0 be a sequence of positive real numbers such that n → 0 with n 2 n → ∞. The sequence of posterior distributions {Π n,θ (• | X n ) : n ≥ 1} is said to contracts around θ 0 at rate n if there is a constant M > 0 such that P (n) [START_REF] Doob | Application of the theory of martingales[END_REF] proposes a general theorem on posterior consistency, which holds under very weak conditions. For a modern exposition of Doob's theorem, we refer to Le Cam and Yang (2000, section 8.2).

θ 0 Π n,θ (θ : d n (θ, θ 0 ) ≥ M n | X n ) → 0.
Theorem 1.2 -Doob's theorem. -Let X be an euclidean space endowed with Borel σalgebra B. Assume that the observations X 1 , . . . , X n ∈ X are identically and independently distributed with distribution P θ 0 on (X , B), and the map θ → P θ is injective. Also assume that (Θ, Σ Θ ) is a Borel set in a complete separable metric space with canonical topology T Θ . Then the sequence of posterior distributions is consistent for T Θ at Π θ -almost all θ 0 .

When the parameter space Θ is finite dimensional, there are finer results about the behavior of {Π n,θ (• | X n ) : n ≥ 1}, such as Bernstein-von Mises theorem (Le [START_REF] Cam | Asymptotics in statistics: some basic concepts[END_REF] which provide sufficient conditions on the model to ensure the asymptotic normality of the sequence of posterior distributions.

In nonparametric Bayes, even consistency can fails, despite theorem 1.2. In fact, Doob's theorem guarantees the posterior consistency up to null sets of the prior distribution. In infinite dimensional models, it is easy to find examples where null sets of the prior distribution are large in some topological sense (see sections 2.4.2 and 2.4.3 in Hjort et al. (2010), and also [START_REF] Freedman | On the asymptotic behavior of bayes' estimates in the discrete case[END_REF]). If the posterior distribution has an explicit expression, it may be possible to prove consistency or rates of convergence by simple Chebychev-type inequalities. However, examples with explicit expression of the posterior distribution are very special and are not to be expected in all applications. The Schwartz-Ghosal-van-der-Vaart theory proposes a general approach to consistency and rates of convergence.

The Schwartz-Ghosal-van-der-Vaart theory

In nonparametric Bayes, sufficient conditions on the model and prior distribution for posterior consistency have been stated by [START_REF] Schwartz | On bayes procedures[END_REF]. Her approach has been extended in the seminal papers of Ghosal et al. (2000); [START_REF]Convergence rates for density estimation with bernstein polynomials[END_REF]Van Der Vaart (2001, 2007a,b) and this to derive posterior concentration rates. Here we present the main ideas behind their approach to posterior rates of convergence. It should be noticed, however, that the conditions on the model and on the prior are less stringent if one seeks only a remains to show that P θ 0 (A c n ) → 0, which is the case if Π(B n (θ 0 , n )) ≥ C 2 exp(-C 1 n 2 n ) as proved in Ghosal and Van Der Vaart (2007a, lemma 10).

A common approach to construct the tests φ n is to find test functions ϕ θ 1 n for testing balls, i.e. H 0 : θ = θ 0 against H 1 : d n (θ, θ 1 ) ≤ r n for some 0 < r < 1 and d n (θ 1 , θ 0 ) ≥ n , that satisfy

P (n) θ 0 ϕ θ 1 n ≤ e -3C 1 n 2 n , sup θ : dn(θ,θ 1 )≤r n P (n) θ (1 -ϕ θ 1 n ) ≤ e -3C 1 n 2 n . (1.2)
For instance, in density estimation existence of such tests for the Hellinger distance and L 1 distance follows from Le Cam (1986, lemma 4, p. 478), whereas in Gaussian regression (see also chapters 2 and 3) this follows from Birgé (2006, proposition 4) for the • 2,n distance. If there exists a finite covering of Θ with N (r n , d n , Θ) exp(2C 1 n 2 n ) balls of radius r n and centers {θ i : 1 ≤ i ≤ N (r n , d n , Θ)}, then φ n can be constructed as

φ n = max ϕ θ i n : d(θ 0 , θ i ) ≥ n and 1 ≤ i ≤ N (r n , d n , Θ) .
If Θ cannot be covered with N (r n , d n , Θ) exp(2C 1 n 2 n ) balls of radius r n , it is enough to find Θ n ⊂ Θ such that N (r n , d n , Θ n ) exp(2C 1 n 2 n ) and P

(n)

θ 0 Π(Θ\Θ n | X n ) → 0.
Then we have the following theorem.

Theorem 1.3 -Let C 1 , C 2 > 0 as previously, 0 < r < 1, and n → 0 with n 2 n → ∞. Suppose that Θ n ⊂ Θ is such that Π(Θ\Θ n ) ≤ exp(-3C 1 n 2 n ) and log N (r n , d n , Θ n ) ≤ 2C 1 n 2
n . Furthermore assume that there is a sequence of tests (ϕ θ 1 n ) n≥1 satisfying equation (1.2), and assume that Π(B n (θ 0 , n )) ≥ C 2 exp(-C 1 n 2 n ). Then P (n)

θ 0 Π(d n (θ, θ 0 ) ≥ n | X n ) → 0.
The condition Π(Θ\Θ n ) ≤ exp(-3C 1 n 2 n ) is sufficient to ensure that P (n) θ 0 Π(Θ\Θ n | X n ) → 0; this follows from a similar argument that we used in the sketch of the proof of proposition 1.1.

Minimax theory

A classical way to assess the asymptotic performances of nonparametric frequentists estimators is to compare the risk they achieve over a class Θ 0 ⊂ Θ to the optimal risk, the so-called minimax risk. Following [START_REF] Tsybakov | Introduction to nonparametric estimation[END_REF], given a semimetric d n , we let the performance of an estimator θ n of θ 0 be measured by the maximum risk of this estimator on Θ 0 : sup θ∈Θ 0 P

(n) θ [d n ( θ n , θ) 2 ].
Then we have the following definition of the minimax risk and the optimal rates of convergence over Θ 0 , respectively.

(n) θ [d n ( θ n , θ) 2 ]
, where the infinimum is taken over all the estimators. Definition 1.5 -Optimal rate of convergence. -A positive sequence ( n ) n≥1 is called an optimal rate of convergence of estimators on (Θ 0 , d n ) if there are constants

C 1 , C 2 > 0 such that lim sup n -2 n R * n ≤ C 1 and lim inf n -2 n R * n ≥ C 2 . An estimator θ n satisfying sup θ∈Θ 0 P (n) θ [d n ( θ n , θ) 2 ] ≤ C 3
Regarding Bayesian procedures, it is proved in Ghosal et al. (2000, theorem 2.5) that if a sequence of posterior distributions {Π(• | X n ) : n ≥ 1} converges at rate n at all θ ∈ Θ 0 for the semimetric d n -under weak conditions on d n -then there exists at least one Bayesian point estimator based on Π(• | X n ) that achieve the maximum risk n (up to constants). In other words, the rate of contraction of {Π(• | X n ) : n ≥ 1} over (Θ 0 , d n ) cannot be faster than the frequentist optimal rate of convergence for (Θ 0 , d n ), otherwise it would imply the existence of a point estimator performing better than the best possible estimator.

Often, the parameter is assumed to belong to a class Θ 0 ≡ A(β, L) where β relates to a notion of smoothness or sparsity and L controls the "size" of the class in a certain sense. As we in general do not know in advance the value of β for θ 0 , we want procedures that achieve optimal rates simultaneously for all value of β; such an estimator is called adaptive over A(β, L).

Nonparametric mixtures

So far, we have introduced the Bayesian paradigm from a general angle. The object of study of this thesis is essentially concerned with the case when the parameter of interest to be estimated is a function from R d to R (or C). Popular prior distributions over function spaces are Gaussian processes [START_REF] Rasmussen | Gaussian processes in machine learning[END_REF], and in the special situation when θ is a probability density function, Dirichlet Process Mixtures [START_REF] Antoniak | Mixtures of dirichlet processes with applications to bayesian nonparametric problems[END_REF][START_REF] Escobar | Bayesian density estimation and inference using mixtures[END_REF].

The use of mixtures has received much less attention outside the density estimation framework, apart from Abramovich et al. (2000); Wolpert et al. (2011); De Jonge and Van Zanten (2010). Before clarifying the meaning of a mixture, we need the following definition of a random measure. We will consider random probability measures and random (positive) measures as particular cases of the more general random signed measures.

Definition 1.6 -Random signed measure. -Let (Ω, E, P) be a probability space and (G, Σ) be a measurable space. We call a mapping Q : Ω × Σ → R ∪ {±∞} a random signed measure if ω → Q(ω, A) is a random variable for each A ∈ Σ and if A → Q(ω, A) is a signed measure for each ω ∈ Ω.

From now, without explicit mention, measures refer to signed measures. Random measures may be either viewed as stochastic processes indexed by a σ-algebra {Q(•, A) : A ∈ Σ}, or as a probability distribution over the set of measures. The latter interpretation is very popular in Bayesian nonparametrics, especially for random probability measures (Ferguson, 1973;[START_REF] Lijoi | Models beyond the dirichlet process[END_REF]. From random measures, it is rather easy to construct random functions, which we call mixtures. and E|X f | ≤ G |f | d|µ|. Now for any f ∈ L 1 (|µ|) we can construct a sequence (f n ) of simple functions such that f n → f pointwise and in L 1 (|µ|) with |f n | ≤ |f |. It follows for any n, m ∈ N that E|X fn -X fm | ≤ G |f n -f m | d|µ|. Hence (X fn ) is Cauchy in L 1 (P), by completeness the limit X f exists and is measurable. Moreover, X f does not depend on the approximating sequence (f n ), i.e. ω → f (x) Q(ω, dx) is measurable and belongs to L 1 (P). Therefore, definition 1.7 is at least satisfactory if the mean measure µ of Q exists and K(•; y) ∈ L 1 (|µ|) for all y ∈ R d ; otherwise we should pay attention to the measurability of ω → θ(ω, y).

Dirichlet Process Mixtures

Since Ferguson (1973), Dirichlet Processes are very popular in Bayesian nonparametrics, with a use across a wide variety of a applications, such as model validation, clustering or density estimation. The Dirichlet Process is a probability distribution over the set of probability measures, defined as follows.

Definition 1.8 -Dirichlet Process. -Let α be a non null finite positive measure on (G, Σ). We say that a random measure P : Ω × Σ → [0, 1] follows a Dirichlet Process with base measure α, abbreviated DP(α), if for all k ∈ N * and all measurable partition (B 1 , . . . , B k ) of G (P (B 1 ), . . . , P (B k )) ∼ Dirichlet(α(B 1 ), . . . , α(B k )).

Beside this formal definition, Dirichlet Processes can be characterized in a series of different ways:

• Let X 1 ∼ α(•)/α(G); and for any k = 1, . . . , n -1 let X k+1 ∼ α k (•)/α k (G), with α k (•) = α(•) + k i=1 δ X i . Then {X i : i ≥ 1} is a Pólya urn sequence with base distribution α. The sequence {X i : i ≥ 1} is exchangeable. Therefore deFinetti's theorem (Kallenberg, 2006, theorem 9.6) implies that there is a random probability distribution P such that conditionally on P , the random variable X 1 , X 2 , . . . are independent. It can be shown that P follows a DP(α) [START_REF] Blackwell | Ferguson distributions via pólya urn schemes[END_REF].

• The stick-breaking construction, due to [START_REF] Sethuraman | A constructive definition of dirichlet priors[END_REF] is the following. Let α := α(G) and V 1 , V 2 , . . . be identically and independently distributed (iid) with distribution Beta(1, α). Set W 1 = V 1 , and for all k ≥ 2 set

W k = V k k-1
j=1 (1 -V j ). Finally let X 1 , X 2 , . . . be iid with distribution α(•)/α, independently of the V k 's. Then P = ∞ k=1 W k δ X k converges almost-surely toward a DP(α) distributed random measure. Note that this construction emphasize the almost-sure discreteness character of the DP.

• If Q : Ω×Σ → R + is a (nonsymmetric) Gamma random measure with base measure α (see below), then P : Ω × Σ → [0, 1] such that P (A) := Q(A)/Q(G) for all A ∈ Σ follows a DP(α) (Ferguson, 1973).

weakened when P ∼ DP(α). Indeed, from Feigin and Tweedie (1989, theorem 4), it suffices to have log(1 + |K(x; y)|) α(dx) < +∞ for all y ∈ R d .

Symmetric Gamma Process Mixtures

A common theme of this dissertation is the use of mixture models outside the popular framework of density estimation with Dirichlet Process Mixtures, in particular in direct or inverse regression. In that case, Dirichlet Processes are no longer a natural choice of random measures, and we shall in general need random signed measures. Lévy Random Measures (LRM) 1 (Wolpert et al., 2011), seem then to be an ideal choice for our purpose, since they are easy to construct and amenable to posterior computations. However, the study of posterior distributions for general LRM is still challenging, so that we restrict ourselves to symmetric Gamma random measures, defined as follows.

Definition 1.9 -Symmetric Gamma random measure. -Let α be a non null finite positive measure on (G, Σ) and η > 0. We say that a random signed measure Q : Ω × Σ → R is a symmetric Gamma random measure with base measure α and scale parameter η if, for all k ∈ N * and all disjoint measurable sets A 1 , . . . , A k ∈ Σ the random variables Q(A 1 ), . . . , Q(A k ) are independent and distributed as the difference of two independent Gamma(α(A i ), η) random variables for all i = 1, . . . , k.

Remark 1.1 -In some problems, it is not worth having recourse to symmetrized random measure; for instance in chapter 4. Obviously, when we are in such situation, we should rely on (nonsymmetric) Gamma random measures, with obvious definition. Moreover, all the results of this section translates to the Gamma process in a straightforward fashion. Now we devote the following sections to constructive definitions of symmetric Gamma random measures and symmetric Gamma process mixtures in terms of the Poisson Process.

Poisson random measures

For a general treatment of Poisson random measures, we refer to Çınlar (2011); Kingman (1992). Let (E, U) be a measurable, locally compact and separable space. Let ν be a σ-finite positive measure on (E, U). A random measure N : Ω × U → N ∪{+∞} is a Poisson random measure with mean ν if (1) For each A ∈ U, N (•, A) ∼ Po(ν(A)) (2) If A 1 , . . . , A n are pairwise disjoint sets in U then N (•, A 1 ), . . . , N (•, A n ) are independent Poisson distributed random variables.

If ν is a σ-finite but not finite measure on (E, U) the above definition still makes sense (because the definition below is consistent with the infinite limit of the Poisson law) if we define,

(3) For all sets A ∈ U with ν(A) = +∞, then N (•, A) = +∞ almost surely.

Let N be a PRM with mean ν and A ∈ U a Borel set, the characteristic function of the random variable N (•, A) is,

E[e it N (•,A) ] = exp -ν(A)(1 -e it ) .

(1.3)

When ν(E) < +∞, there is a convenient way to construct and interpret PRM. Indeed, start with the probability measure π(•) = ν(•)/ν(E) on (E, U) and let K ∼ Po(ν(E)), X k iid ∼ π(•) for 1 ≤ k ≤ K. From this sample form the following measure,

N (A) = K i=1 1 A (X k ), A ∈ U.
(1.4) An easy computation shows that the characteristic function of equation (1.4) is nothing else but equation (1.3) and thanks to the uniqueness of the Fourier transform, the random measure defined in equation (1.4) is a PRM with mean ν. When ν(E) = +∞ we can find a disjoint partition {E i } ∞

i=1 of E such that ν(E i ) < +∞ for all i = 1, . . . , ∞. Now define N i the PRM with mean ν(E i ) on the subset E i . By the above discussion N i is almost-surely purely atomic. Let N (A) = ∞ i=1 N i (A ∩ E i ) for all A ∈ U. As N i (A ∩ E i ) are independent Poisson random variables with means ν(E i ), it follows that N (A) is a Poisson random variable with mean ν(E) = +∞ and hence N is a Poisson random measure with mean ν. As a consequence we have the following proposition.

Proposition 1.2 -Let ν be a σ-finite measure on (E, U) and N a PRM with mean ν. Then,

(1) N is almost surely purely atomic.

(2) If ν(E) < +∞, then N as almost-surely a finite number of atoms.

(3) If ν(E) = +∞, then N as almost-surely a countably infinite number of atoms, but for all compact A ∈ U with ν(A) < +∞, N has almost-surely a finite number of atoms in A.

Poisson representation of symmetric Gamma process mixtures

Let η > 0, H η (du) := |u| -1 e -|u|η du, α be a non null finite positive measure on (G, Σ) and N be the PRM with mean H η × α on (R × G, B R ⊗ G). Then (see for instance Wolpert et al. (2011)) the random measure Q such that

Q(•, A) := R×A u N (•, dudx), A ∈ Σ
has the distribution of a symmetric Gamma random measure with base measure α and scale parameter η. Moreover, given a jointly measurable kernel K : G × R d → R, the mixture of K by Q can be represented as

θ(•, y) = R×G uK(x; y) N (•, dudx) ≡ ∞ i=1 u i K(x i ; y),
where (u i , x i ) ∞ i=1 are the atoms of a Poisson random measure with mean H η × α; the series has almost-surely an infinite number of terms, and converges (pointwise) almostsurely if R×G min(1, |uK(x; y)|)H η (du)α(dx) < +∞ for all y ∈ R d . For other modes of convergence, see Wolpert et al. (2011, section 4).

Other representations and connections

Just to mention a few, other representations of the symmetric Gamma random measures, based on the Poisson process, may be found on [START_REF] Rosiński | Series representations of lévy processes from the perspective of point processes[END_REF].

Also, it is well known (see for instance Tsilevich et al., 2000, lemma 2) that if Q : Ω × Σ → R + is a (nonsymmetric) Gamma random measure with base measure α on (G, Σ) and scale parameter η > 0, then the random variable ω → Q(ω, G) and the random variables ω → Q(ω, A)/Q(ω, G) for all A ∈ Σ are independent. Hence, the construction of the Dirichlet random measure from Gamma random measures can be easily inverted, allowing to represent Gamma random measures in term of the Dirichlet Process.

The latter connection between Dirichlet Process and Gamma random measures is exploited in section 1.3.2 below to design an algorithm for simulating the posterior of Gamma process mixtures, and in a lesser extent has been one of the motivation of the algorithm proposed in chapter 2.

Concrete examples of mixtures

So far we discussed nonparametric mixture models in all their generality. Here we present some concrete examples with related existing asymptotic results.

In the sequel for any random measure

Q : Ω × Σ → R ∪ {±∞}, Q(A) is understood as Q(•, A).
Similarly, we make implicit the dependence on ω ∈ Ω of all functional of Q.

Location and Location-scale mixtures

Given a measurable mother function g : R d → R, we define the location-scale kernel K(A, µ; y) ≡ K A (x -µ) := det(A) -1/p g(A -1 (x -µ)) for all x ∈ R d and all A ∈ E, where E is the set of all d × d positive define real matrices; it may be convenient to adapt the power factor 1/p depending on the statistical model considered. In probability density estimation, g is a probability density and p = 1, while this constraint is not necessary in regression.

We recall the following definitions of a location mixture and, respectively, of a locationscale mixture prior.

Definition 1.10 -Let E 0 ⊂ E and R d be endowed with Borel σ-algebra. Let Π A be a prior on E 0 , and

Q : Ω × B(R d ) → R ∪ {±∞} be a random measure. Then θ(x) := K A (x -µ) Q(dµ), where A ∼ Π A , is a location mixture of g by Q (provided it is a well-defined random variable at all x ∈ R d ). Definition 1.11 -Let E 0 ⊂ E and let E 0 × R d be endowed with Borel σ-algebra. Let Q : Ω × B(E 0 × R d ) → R ∪ {±∞} be a random measure. Then θ(x) := K A (x -µ) Q(dAdµ) is a location-scale mixture of g by Q (provided it is a well-defined random variable at all x ∈ R d ).
terior consistency at θ 0 when θ 0 (x) = K σ 0 (x-µ) dP 0 (µ) with some compactly supported probability measure P 0 , or θ 0 is compactly supported and lim σ→0 θ 0 log(θ 0 /K σ * θ 0 ) = 0.

In the meantime, still in univariate density estimation, [START_REF] Genovese | Rates of convergence for the gaussian mixture sieve[END_REF] provided a first attempt to rates of convergence for location-scale mixtures of Gaussians, although with a frequentist sieve estimator θ n . Under the assumption that 1), where d is the Hellinger distance. The rates t n crucially depend on the tails of P 0 ; the best rate is achieved in the compactly-supported case with t n = (log n/n) 1/4 . Their rate is clearly suboptimal, as we expect to achieve a nearly parametric rate of n -1/2 (up to a power of log n) in this supersmooth situation. Ghosal and Van Der Vaart (2001) improved the preceding result and recovered a rate of (log n) κ / √ n, where κ depends on the tails of P 0 . Also, assuming that P 0 has sub-Gaussian tails, and that the bandwidth σ 0 is bounded from above and below, they obtained sufficient conditions to achieve the posterior contraction rate of (log n) κ / √ n for the Hellinger distance with a Dirichlet process location mixture of Gaussians prior. [START_REF] Tokdar | Posterior consistency of dirichlet location-scale mixture of normals in density estimation and regression[END_REF] established weak and L 1 consistency for Dirichlet process location-scale mixtures of Gaussians (again in univariate density estimation) under mild assumptions on θ 0 . Particularly, θ 0 is not assumed to be itself a mixture and can be heavy-tailed.

θ 0 (x) = K σ (x -µ) dP 0 (µ), they found (t n ) n≥1 such that P n θ 0 (d(θ 0 , θ n ) > t n ) = o(
Considering univariate density estimation with Dirichlet process location mixtures, assuming θ 0 twice continuously differentiable with compact support, Ghosal and Van Der Vaart (2007b) established a nearly minimax contraction rate of n -2/5 (up to a power of log n) for the Hellinger distance. This is the first time that posterior rates are established without assuming θ 0 supersmooth. However, if θ 0 is more than twice continuously differentiable, their rate does not improve and becomes suboptimal.

The main difficulty in improving the previous rates rely on the ability of approximating a smooth density θ 0 with a convex mixture of densities, when θ 0 is not itself a mixture 2 . With the assumption log θ 0 ∈ C β , Kruijer et al. (2010) used an idea from [START_REF] Rousseau | Rates of convergence for the posterior distributions of mixtures of betas and adaptive nonparametric estimation of the density[END_REF] to circumvent the issue, leading the way to rate adaptive posterior contraction rates of n -2β/(2β+1) (up to a power of log n) over the whole classes of β-log-Hölder densities with exponential tails, β > 0, using Dirichlet process location mixtures of generalized Gaussians and either Hellinger or L 1 distance.

The first asymptotic result for multivariate location mixtures in regression appeared in De Jonge and Van Zanten (2010). The authors considered the Gaussian errors regression with fixed design on [0, 1] d for a finite mixture model, although with a random number of components. Their result is near optimal and rate adaptive over the Hölder classes C β , β > 0, for the • 2,n distance, and holds for a quite general family of mother functions. [START_REF] Scricciolo | Posterior rates of convergence for dirichlet mixtures of exponential power densities[END_REF] investigated univariate density estimation with Dirichlet process mixtures of exponential power densities, assuming that the true density is itself a mixture with mixing measure having sub-exponential tails. She found nearly parametric posterior rates of convergence when the mother function is differentiable at 0, and slower rates otherwise.

Regarding multivariate density estimation with Dirichlet process location mixtures of Gaussians, Shen et al. (2013) established near optimal and adaptive rates over anisotropic Hölder classes for the Hellinger distance, provided exponential tails assumptions on θ 0 .

Their approach allowed to release the assumption log θ 0 ∈ C β used in Kruijer et al. (2010) onto θ 0 ∈ C β , provided supplementary conditions on ratio of the form D k f /f p . Canale and De Blasi (2017) extended the previous result to multivariate density estation with Dirichlet process location-scale mixtures of Gaussians. They obtained suboptimal rates for the Hellinger distance over the Hölder classes, assuming exponential tails for the true density. Moreover, they conjectured that under weaker tail assumptions their bound on the rate remains true and should be near optimal. We believe today that this conjecture is wrong (see below).

Recently, Scricciolo (2014) investigated univariate density estimation with Pitman-Yor process and normalized Inverse-Gamma process location mixtures. The novelty here is that she looked at contraction rates under L p metric, for 1 ≤ p ≤ ∞. For a class of analytic functions, she found that the procedure is nearly optimal, provided θ 0 has exponential tails. However, regarding at β-Sobolev classes, β < ∞, the rates are optimal only when 1 ≤ p ≤ 2, and deteriorate by a genuine power of n otherwise.

Some questions remain open about location and location-scale mixtures. Why do the rates found in Scricciolo (2014) for the L p distance deteriorate when p ≥ 2 and the true function is not analytic ? We even don't know if there is a Bayesian procedure that achieve the optimal rates in this situation [START_REF] Giné | Rates of contraction for posterior distributions in l r-metrics, 1 r[END_REF].

Another interesting issue concerns the exponential tail assumption on θ 0 . Currently it appears difficult to get rates of contraction for density estimation with location or location-scale mixture without light tail assumption on θ 0 . Under light-tail hypothesis, everything happens as if θ 0 is compactly supported, so that it is not surprising that location mixtures perform better than location-scale mixtures (Shen et al., 2013;Canale and De Blasi, 2017). It is known from the frequentist litterature [START_REF] Juditsky | On minimax density estimation on R[END_REF]Reynaud-Bouret et al., 2011;Goldenshluger and Lepski, 2014) that without tails assumptions, the optimal rates for β-Hölder classes, for the L p distance, deteriorate when 1 ≤ p ≤ 2 + 1/β in comparison to the optimal rates under tail condition. For p = 1, and hence for the Hellinger distance, the rates deteriorate to 2 n = 1 if no tails assumptions are made, and may take a bunch of values between 1 and n -2β/(2β+1) depending on the heaviness of the tail. This explains why Canale and De Blasi (2017) conjectured that the suboptimal rates they found with location-scale mixtures of Gaussians under light tail assumptions may become optimal if they could release the tail condition. We do not think that this conjecture is correct (see also chapter 3), and we present here an argument to explain why we think the conjecture is false.

Assume d = 1 for simplicity. Define the pointwise Hölder regularity as follows. Let θ : R → R be a function, α > 0, α / ∈ N and x 0 ∈ R. Then θ ∈ C α (x 0 ) if and only if there exists a real η > 0, a polynomial P with degree less than α and a constant

C > 0 such that |f (x) -P (x -x 0 )| ≤ C|x -x 0 | α for all |x -x 0 | ≤ η. The pointwise Hölder exponent of θ at x 0 is β(x 0 ) = sup{α : θ ∈ C α (x 0 )}.
Returning to (global) Hölder spaces C β , there are functions for which β(x) = β for all x ∈ R. This seemlingly innocuous fact is indeed crucial, because for these functions, there is no reason for location-scale mixtures to perform better order of approximation than location mixtures. One could argue that there is no need to approximate θ 0 with the same order of approximation in zones with few data as in zone of dense data, encouraging location-scale mixtures. This argument is hard to implement in density estimation, but could be rather easily applied in regression with Gaussian errors (see chapter 3), still leading to suboptimal rates for location-scale mixtures.

In chapter 3 for random design regression, we obtain rates for both location and location-scale mixtures under various tail constraints, which are always worst for locationscale mixtures than for location mixtures. The novelty here is that, under heavy tail assumptions, both location and location-scale appear to be suboptimal and we can find hybrid procedure that performs always better than both types of mixtures. That said, to be able to accept or reject definitively Canale and De Blasi's conjecture, it is necessary to establish lower bounds on the rate of contraction of the posterior (in the sense of [START_REF] Castillo | Lower bounds for posterior rates with gaussian process priors[END_REF]), which remains a challenge for mixture models.

Finally, we believe that although location-scale mixtures are suboptimal over Hölder classes, they should outperform location mixtures if looking at classes of locally varying smoothness, making them useful in practice.

Mixtures of Betas and Gammas

Another common examples are mixtures of Beta densities with respect to a random probability measure. Although this prior makes sense only in density estimation over [0, 1], we would like to mention a few asymptotic results, in particular because the tools developed in the next references have been useful in the study of posterior distribution for general mixtures.

Consistency for Dirichlet process mixtures of Bernstein polynomials goes back to [START_REF] Petrone | Consistency of bernstein polynomial posteriors[END_REF] under very mild assumptions on the true density. For some prior distributions, [START_REF]Convergence rates for density estimation with bernstein polynomials[END_REF] established nearly parametric rates of convergence for the Hellinger distance under the assumption that the true density is itself of mixture of Bernstein polynomials, and provided suboptimal rates otherwise.

Regarding at Dirichlet process mixtures of Beta densities, [START_REF] Kruijer | Posterior convergence rates for dirichlet mixtures of beta densities[END_REF] established posterior rates over β-Hölder classes for 0 < β ≤ 2, in Hellinger distance, achieving optimal adaptive rate over 0 < β ≤ 1, and suboptimal otherwise. For the L 1 distance, [START_REF] Rousseau | Rates of convergence for the posterior distributions of mixtures of betas and adaptive nonparametric estimation of the density[END_REF] is the first article in which mixture priors are shown to yield nearly optimal and adaptive rates over the whole β-Hölder classes for any β > 0, paving the way for the results of De Jonge and Van Zanten (2010); Kruijer et al. (2010); Shen et al. (2013) in location mixtures.

Recently, Bochkina and Rousseau (2016) investigated posterior contraction rates for density estimation on the positive real line based on mixtures of Gamma distributions. They required less stringent conditions on the tails of the true density than the usual exponential tail condition required for location and location-scale mixtures. Together with the result of Canale and De Blasi (2017), this paper was a supplementary motivation for the work presented in chapter 3.

Contributions

This section presents the main contribution of this manuscript to theoretical results about symmetric Gamma process mixtures.

Fixed design regression with symmetric Gamma process mixtures

In chapter 2, we are concerned with the problem of a random response Y corresponding to a deterministic covariate vector x taking values in [-S, S] d for some S > 0. We aim at estimating the regression function f : [-S, S] d → R such that f (x i ) = E Y i , based on independent observations of Y . More precisely, the nonparametric regression model we consider is the following,

Y i | i = f (x i ) + i , i = 1, . . . , n, 1 , . . . , n |σ 2 i.i.d ∼ N (0, σ 2 ), independently of (f, σ), (f, σ) ∼ Π,
with Π the distribution on an abstract space Θ, given by σ ∼ P σ independently of f drawn from the distribution of a symmetric Gamma process mixture.

We consider two families of mixtures. First, the location-scale mixtures, as defined in definition 1.11; and second we introduce a new type of mixtures, which we call locationmodulation mixtures, defined as follow.

For a measurable mother function g : R d → R, we define the location-modulation

kernel K ξ,φ (x) := g(x) cos( d i=1 ξ i x i + φ), for all x ∈ R d , all ξ ∈ R d and all φ ∈ [0, π/2]. Then, a mixture of K ξ,φ by a symmetric Gamma random measure Q : Ω × B(R d × R d × [0, π/2]) → [-∞, ∞]
is the distribution of the random function (for those g for which it is well-defined)

f (•, x) := R d × R d ×[0,π/2] K ξ,φ (x -µ) Q(•, dξdµdφ), ∀x ∈ R d .
Our main result concerns the posterior rates of convergence for the two families of prior distributions. Assuming that the true regression function f 0 belongs to the Hölder space C β [-S, S] d , we find that the parameters of the mixing symmetric Gamma random measure and g can be chosen so that the following posterior rates of convergence hold for these models, with the empirical metric

d n ((f 1 , σ 1 ), (f 2 , σ 2 )) := (n -1 n i=1 |f 1 (x i ) - f 2 (x i )| 2 ) 1/2 + | log σ 1 -log σ 2 |:
• Location-scale mixtures: there is t > 0 such that 2 n n -β/(2β+d+κ/2) (log n) t , where κ > 0 depends on the parameters of the prior.

• Location-modulation mixtures: there is t > 0 such that

2 n n -β/(2β+d) (log n) t .
Regarding location-scale mixtures, the same phenomena as in density estimation with DPM take place here (Canale and De Blasi, 2017), yielding to suboptimal rates. However, it seems like location-modulation behave better over the β-Hölder classes as they achieve the minimax rate of convergence (up to a power of log n factor). univariate regression models:

Y i = f (X i ) + i , i i.i.d ∼ N (0, s 2 ), i = 1, . . . , n, X 1 , . . . , X n i.i.d ∼ Q 0 , f ∈ L 2 (Q 0 ).
We assume that s is known, which is just a matter of convenience for proofs, and we wish to estimate the parameter f using location or location-scale mixtures of Gaussians by symmetric Gamma random measures. Our aim is to study posterior concentration rates around the true regression function f 0 under the empirical 2 distance of the covariates, defined as

d n (f, f 0 ) 2 := n -1 n i=1 |f (x i ) -f 0 (x i )| 2 .
We assume that f 0 ∈ L 1 and belongs to a Hölder ball with smoothness β. The tail condition are then on the design distribution and written as R |x| p dQ 0 (x) < +∞, p ≥ 0.

We show in chapter 3, that in most cases location mixtures have a better posterior concentration rate than location-scale mixtures and unless p goes to infinity the posterior concentration rates is not as good as the usual n -β/(2β+1) . This rate is suboptimal for light tail design points, since in this case the minimax posterior concentration rate is given by n -β/(2β+1) . To improve on this rate we propose a new version of location-scale mixture models, which we call the hybrid location-scale mixture and we show that this nonparametric mixture model leads to better posterior concentration rates than the location mixture (and thus than the location-scale mixture). All these results are up to log n terms. The results are summarized in table 1.1 which displays the value q defined by 2 n = n -q .

Table 1.1: Summary of posterior rates of convergence for different types of mixtures. The rates are understood to be in the form 2 n = n -q , up to powers of log n factors, where q is given below.

0 < p < 2 p ≥ 2 0 < p ≤ 2β β + 1 2β β + 1 < p ≤ 2β p > 2β 0 < p ≤ 2β β + 1 2β β + 1 < p ≤ 2β p > 2β Location 2β 3β + 1 2β 3β + 1 2β 3β + 1 2β 2β + 1 + 2β/p 2β 2β + 1 + 2β/p 2β 2β + 1 + 2β/p Location-scale 2β 3β + 2 2β 2β + 1 + 2β/p 2β 2β + 2 2β 3β + 2 2β 2β + 1 + 2β/p 2β 2β + 2 Hybrid 2β 3β + 1 p p + 1 2β 2β + 1 2β 3β + 1 p p + 1 2β 2β + 1
Although the results are presented in the regression model, we believe that similar phenomena should take place in the density estimation problem.

Finally, we also compute the rates for location mixtures when p = 0 and the parameters of the mixing symmetric Gamma random measures are dependent of the covariates. The strategy can be easily translated to location-scale and hybrid mixtures.

Posterior sampling

Quite generally, in Bayesian nonparametrics, the posterior distribution is either analytically intractable, or difficult to compute. A popular way to circumvent the issue is to construct a Markov chain whose transition kernel admits the posterior distribution as invariant distribution; such algorithms are called Markov Chain Monte Carlo algorithms (MCMC). Then from the samples of the Markov chain, one can (approximately) evaluate all functionals of the posterior distribution.

Although Dirichlet Process Mixtures (DPM) is not the object of this dissertation is, we first review an algorithm for sampling the posterior distribution of DPM (in density estimation model); the interest is two fold. First, it will highlight the difficulties encountered for sampling mixture models in general models. Second, the algorithm presented in the next section has been a great source of inspiration for the design of the algorithm proposed in chapter 2 (because of the close relationship between Dirichlet Processes and Gamma Processes).

We will conclude this section with an overview of the problems encountered with general mixture models, followed by a presentation of previsouly existing algorithm for posterior simulation, which could be used as alternative to the algorithm of chapter 2.

Gibbs sampling Dirichlet Process Mixtures

Gibbs methods for sampling the posterior distribution of DPM go back to Escobar's unpublished thesis; published later in [START_REF] Escobar | Estimating normal means with a dirichlet process prior[END_REF]. The emergence of these algorithms signed the boom of Bayesian nonparametrics, which had a solid theoretical framework for a while, but lacked to be used effectively in practice. Gibbs algorithms for sampling the posterior of DPM may be classified in two categories.

• Conditional samplers. These algorithms use an explicit representation of the measure P ∼ DP(α), mainly the stick-breaking representation [START_REF] Ishwaran | Gibbs sampling methods for stick-breaking priors[END_REF][START_REF] Papaspiliopoulos | Retrospective markov chain monte carlo methods for dirichlet process hierarchical models[END_REF][START_REF] Walker | Sampling the dirichlet mixture model with slices[END_REF][START_REF] Kalli | Slice sampling mixture models[END_REF]. We will not review these methods here. For an exhaustive review (among other cool Bayesian nonparametrics stuff) we recommend the unpublished thesis of [START_REF] Fall | Modélisation stochastique de processus pharmaco-cinétiques, application à la reconstruction tomographique par émission de positrons (TEP) spatio-temporelle[END_REF]. • Collapsed samplers. These algorithms marginalize the distribution of P ∼ DP(α) in the posterior distribution, and are the first techniques developed for DPM sampling.

Escobar's algorithm falls into that category [START_REF] Escobar | Estimating normal means with a dirichlet process prior[END_REF][START_REF] Escobar | Bayesian density estimation and inference using mixtures[END_REF].

However, Escobar's algorithm suffers from two shortcomings. First, the mixing time of Escobar's algorithm turns out to be rather huge and inefficient. Second, the algorithm works only when the likelihood of the parameter is conjugate to the base distribution α of the mixing Dirichlet Process. The first issue is addressed for instance in [START_REF] Neal | Bayesian mixture modeling[END_REF]; [START_REF] West | Hierarchical priors and mixture models, with application in regression and density estimation[END_REF]; [START_REF] Maceachern | Estimating normal means with a conjugate style dirichlet process prior[END_REF], while the second issue is addressed in MacEachern and [START_REF] Maceachern | Estimating mixture of dirichlet process models[END_REF]; [START_REF] Walker | Sampling methods for bayesian nonparametric inference involving stochastic processes[END_REF]; Neal (2000).

We now review briefly Neal (2000, algorithm 8), which is of particular interest for the comprehension of chapter 2. We let α be a finite positive measure over (G, Σ), α = α(G), and K : G × R d → R + be a jointly measurable kernel such that K(x; •) is a probability density function for all x ∈ G. Neal's algorithm is based on the following hierarchical representation of DPM.

y i |x i ∼ K(x i ; •) ∀i = 1, . . . , n x 1 , . . . , x n |P i.i.d ∼ P P ∼ DP(α).
(1.5)

The samples y 1 , . . . , y n from the model of equation (1.5) are identically distributed with distribution density y → K(x; y) dP (y), where P ∼ DP(α). If we marginalize P in the equation (1.5), then x 1 , . . . , x n has the distribution of a Pólya urn sequence with base measure α (see section 1.2.1); this fact is crucial and is one of the foundation of the algorithm developed later in chapter 2.

Going back to Neal's algorithm and equation (1.5), each sample y i is associated with a component x i of the mixture. However, because of the almost-sure discreteness of the Dirichlet Process, there might be some identical values of x i . We let x 1 , x 2 , . . . denote the unique values of x 1 , . . . , x n , and we introduce the latent clustering variables c 1 , . . . , c n such that

c i = k if x i = x c i = x k . Observation y i with c i = k is said to belong to cluster k.
Neal's algorithm produces a Markov chain whose invariant distribution is the posterior distribution of (c 1 , . . . , c n , x 1 , x 2 , . . . ). The algorithm draws samples by successively sampling from

• c 1 , . . . , c n |y 1 , . . . , y n , x 1 , x 2 , . . . using the auxiliary variables method, as described below; • x 1 , x 2 , . . . |y 1 , . . . , y n , c 1 , . . . , c n using the Metropolis-Hastings algorithm [START_REF] Hastings | Monte carlo sampling methods using markov chains and their applications[END_REF], or any other method which allows to draw samples from this distribution.

We now briefly describe the auxiliary variables method. Assume that you want to draw samples from a distribution π x . Then you can perform the following update scheme for x. First choose a joint distribution π xy for (x, y) that admits π x as marginal. Then draw a sample of y from the conditional distribution of y|x followed by a update of (x, y) that leave π xy invariant. Finally discard y and keep only x. This update for x will keep π x invariant. Neal (2000) uses the auxiliary variable methods to update (c i ) n i=1 . He uses m ∈ N * auxiliary variables representing new potential clusters. The choice of m is governed by a trade-off between computation cost and mixing efficiency.

Each c i is updated using Gibbs sampling. Since the observations (y i ) n i=1 are exchangeable, we can assume we are updating c i for the last observation. We let k -denote the number of clusters that contain at least one observation y i for i = 1, . . . , n -1, and we label these clusters with distinct values in {1, . . . , k -}. The prior probability of c i to be equal to 1 ≤ l ≤ k -is n -i,l /(n -1 + α), where n -i,l denote the number of observations y j with j = i in cluster l. The probability of c i having other value is α/(n -1 + α), which is split among the m auxiliary clusters that are associated with parameters x 1 , . . . , x m drawn independently from α(•)/α. As shown in Neal (2000), this representation of the prior translates to the posterior distribution, yielding the algorithm 1.

Sampling symmetric Gamma process mixtures

When dealing with other statistical models than density estimation with DPM, in general we cannot associate an observation to a unique component of the mixture, as in equation (1.5). In chapter 2, we will introduce a representation of the symmetric Gamma process which allows to adapt algorithm 1 to symmetric Gamma process mixtures. Meanwhile, we present here the two main existing alternatives to the algorithm we propose, taken respectively from [START_REF] Wolpert | A nonparametric bayesian approach to inverse problems[END_REF]Wolpert et al. ( , 2011) ) and [START_REF] Erhardsson | Non-parametric bayesian inference for integrals with respect to an unknown finite measure[END_REF]. The goal of these algorithms is the use of mixtures outside the density estimation framework.

Algorithm 1 Neal (2000, algorithm 8) Let m ∈ N * and the state of the Markov Chain consist of (c 1 , . . . , c n ) and (x 1 , x 2 , . . . ). Repeatedly sample as follows.

• For all i = 1, . . . , n: Let k -denote the number of clusters that contain at least one observation y j for j = i, and k denote the total number of clusters.

-

If c i = c j for all j = i, then k -= k -1 and do * x ← x c i * (x 1 , x 2 , . . . , x c i -1 , x c i +1 , . . . ) ← (x 1 , x 2 , . . . , x c i -1 , x c i , x c i +1 , . . . ) * x k ← x and for all k with c k ≥ c i : c k ← c k -1 * Draw values independently from α(•)/α(G) for those x l for which k + 1 ≤ l ≤ k + m -1.
-Else if there is one j = i such that c i = c j , then k -= k and do * Draw values independently from α(•)/α(G) for those x l for which k

+ 1 ≤ l ≤ k + m.
Draw a new value for c i from {1, . . . , k -+ m} using the following probabilities

P(c i = l | c -i , y i , x 1 , x 2 , . . . ) ∝    n -i,l n-1+α(G) K(x l ; y i ) 1 ≤ l ≤ k - α/m n-1+α(G) K(x l ; y i ) k -< l ≤ h.
Change the state to contain only those x 1 , x 2 , . . . that are associated to at least one observation (relabel the clusters if necessary). Let α be a finite positive measure on (G, Σ), η > 0, > 0 and define H η, (du) :=

1 |uη|> (u)|u| -1 e -|u|η du. Let K : G × R d → R be a jointly measurable kernel such that min(1, |uK(x; y)|) H η,0 (du)α(dx) < +∞,
and N be the Poisson random measure with mean H η, × α. For all y ∈ R d , the mixture θ (•, y) := uK(x; y) N (•, dudx) converges in probability to a mixture of K by a symmetric Gamma random measure with parameters (α, η) as → 0 (Wolpert et al., 2011).

Let E 1 be the exponential integral E 1 function defined by E 1 (x) := ∞ x t -1 e -t dt for all x > 0 and let P ,u be the distribution which has density 1 |uη|>0 (u)|u| -1 e -|u|η /(2E 1 ( ))

with respect to the Lebesgue measure on R. By virtue of section 1.2.2, θ (y) has the almostsure representation θ (y) = J j=1 u j K(x j ; y), where J ∼ Po(2α(G)E 1 ( ))), u j i.i.d

∼ P ,u for j = 1, . . . , J, and x j i.i.d

∼ α(•)/α(G) for all j = 1, . . . , J, independently of (u j ) J j=1 . Using the previous approximation of symmetric Gamma process mixtures, [START_REF] Wolpert | A nonparametric bayesian approach to inverse problems[END_REF]Wolpert et al. ( , 2011) ) implement a reversible jumps MCMC algorithm [START_REF] Green | Reversible jump markov chain monte carlo computation and bayesian model determination[END_REF] with birth-and-death move proposals [START_REF] Cappé | Reversible jump, birth-and-death and more general continuous time markov chain monte carlo samplers[END_REF], which we summarize in algorithm 2 (see also the manuscript thesis of [START_REF] Tu | Nonparametric modelling using Lévy process priors with applications for function estimation, time series modeling and spatio-temporal modeling[END_REF] for complements).

Algorithm 2 Wolpert et al. (2011, Birth-Death algorithm) Let p b , p d ∈ (0, 1) with p b + p d < 1 be parameters of the algorithm.

• At t = 0 initialize the Markov chain at S 0 .

• At iteration t ≥ 1:

-With probability p b : S t ← BIRTH(S t-1 ).

-With probability 1-p b : Select randomly an index j uniformly in {1, . . .

, J t-1 }. * With probability 1 -p b -p d : Draw u * j = u t-1 j + N (0, σ 2 u ). If |u * j η| ≤ then S t ← DEATH(j, S t-1 ). Otherwise S t ← UPDATE(j, u * j , S t-1
). * With probability p d : S t ← DEATH(j, S t-1 ).

We now clarify the notations used in algorithm 2, as well as details of the three BIRTH, DEATH and UPDATE moves involved in the algorithm. We use the vector notations u := (u 1 , . . . , u J ) and x = (x 1 , . . . , x J ). The state of the Markov chain at iteration t is S t = (J t , u t , x t ), corresponding to a function θ t (y) = J t j=1 u t j K(x t i ; y). We denote by L(S t | y n ) the likelihood of S t (equivalently θ t ) under the observations y n = (y 1 , . . . , y n ). We let p b , p d ∈ (0, 1) with p b + p d < 1, and p u = 1 -p b -p d . The main steps involved in the algorithm are the following.

• BIRTH step. Set J * = J t-1 + 1 and draw uniformly a index j in {1, . . . , J * }.

Draw u * j ∼ P ,u and x * j ∼ α(•)/α. It is possible to draw u * j directly from P ,u by numerically inverting the CDF; otherwise draw from other distribution and adapt the Metropolis-Hastings ratio below. Set u * = (u t-1 1 , . . . , u t-1 j-1 , u * j , u t-1 j , . . . ) and x * = (x t-1 1 , . . . , x t-1 j-1 , x * j , x t-1 j , . . . ). With probability 1 ∧ H accept the proposal and set S t = S * ; else reject the proposal and set S t = S t-1 . The Metropolis Hastings ratio H for this move is

H = L(S * | y n ) L(S t-1 | y n ) 2αE 1 ( ) J * p d + p u Φ -u * j + /η σu -Φ -u * j -/η σu /J * p b /J * ,
where Φ is the CDF of the N (0, 1) distribution. • UPDATE step. When this step is implemented, we have at our disposal a random index j and a candidate u * j . We obtain S t after the following sub-steps. -First, we accept the proposal u * j with probability 1∧H, where H is the classical Metropolis Hastings ratio for this move.

-Second, we update x t-1 j for x j using random-walk Metropolis-Hastings.

• DEATH step. Set u * = (u t-1 1 , . . . , u t-1 j-1 , u t-1 j+1 , . . . ) and x * = (x t-1 1 , . . . , x t-1 j-1 , x t-1 j+1 , . . . ). Let J * = J t-1 -1, S * = (J * , u * , x * ). With probability 1∧H we accept the proposal and set S t = S * , otherwise we reject the proposal and set S t = S t-1 . The Metropolis Hastings ratio H for this move is

H = L(S * | y n ) L(S t-1 | y n ) J t-1 2αE 1 ( ) p b /J * p d + p u Φ -u t-1 j + /η σu -Φ -u t-1 j -/η σu /J * .
Erhardsson's importance sampling algorithm [START_REF] Erhardsson | Non-parametric bayesian inference for integrals with respect to an unknown finite measure[END_REF] proposed an importance sampling algorithm for simulation of Gamma process mixtures. His algorithm should be rather straightforwardly symmetrized to handle symmetric Gamma process mixtures.

Broadly, [START_REF] Erhardsson | Non-parametric bayesian inference for integrals with respect to an unknown finite measure[END_REF] used the Markov Chain from Feigin and Tweedie (1989)whose invariant distribution is a Dirichlet Process -in combination with the discussion of section 1.2.2 regarding the representation of Gamma random measures in term of the Dirichlet Process. This yields the algorithm 3, where we use essentially the same notations as in previous section. We should not insist more since the algorithm is relatively inefficient, however, the idea is interesting and might be improved in the light of recent progress made around importance sampling algorithms.

Algorithm 3 [START_REF] Erhardsson | Non-parametric bayesian inference for integrals with respect to an unknown finite measure[END_REF] Importance sampling algorithm

• Initialize G 0 (y) = K(x 0 ; y) for some arbitrary x 0 ∈ G.

• Let {U i : i ≥ 1} iid. with distribution Beta(α(G), 1), {x i : i ≥ 1} iid. and independent of {U i : i ≥ 1} with distribution α(•)/α(G), and independently of everything else {Z i : i ≥ 1} be iid. with distribution Ga(α(G), η).

• For 1 ≤ i ≤ N , let G i (y) = (1 -U i )K(x i ; y) + U i G i-1 (y), and G † i (y) = Z i G i (y). • Compute the importance weights w i = L(G † i | y n ). • Define the mixture X n such that P( X N = G † i | G † 1 , . . . , G † N ) = w i / N i=1
w i for all i = 1, . . . , N . Then X n (y) converges in probability to a θ(y), where θ is a Gamma process mixtures of K with base measure α and scale parameter η.

Contribution to posterior sampling of symmetric Gamma process mixtures

Recall the definition of a Pólya urn sequence from section 1.2.1. We say that a random variable J has a SGa(1, 1) distribution if it is distributed as the difference of two independent Ga(1, 1) random variables. Then, inspired from a paper of Favaro et al. (2012), we prove in chapter 2 the following representation of symmetric Gamma random measures. The convergence of signed measures in the theorem is understood with respect to the weak-* topology on the space of signed Radon measures, identified to the dual space of the space of continuous functions with compact support.

random measure, Q p := T /p p i=1 J i δ X i . Then Q p d → Q
, where Q is a symmetric Gamma random measure with base distribution αF (•) and scale parameter √ η.

We use the previous representation of symmetric Gamma random measures to adapt algorithm 1 for sampling posterior distribution of symmetric Gamma process mixtures, as described in chapter 2.

Quantum Homodyne Tomography

This last section is devoted to the introduction of necessary material for proper understanding of chapter 4. When we studied mixtures as prior distribution in direct regression problems (chapters 2 and 3) we never lost sight of our original belief that mixtures were good priors for solving statistical inverse problems. Chapter 4 constitutes the achievement of our initial goal, we propose a mixture model for estimating the quantum state of a light beam through Quantum Homodyne Tomography measurements. Quantum Homodyne Tomography is a very challenging, nonlinear statistical inverse problem.

Physical background

Quantum mechanics studies the microworld, the physical laws taking place at the microscopic level, that cannot be adequately described by classical mechanics. Unlike classical mechanics, the predictions of quantum mechanics are probabilistic so that we cannot (in general) infer the result of a single measurement, but only the distribution of possible outcomes.

Here we formulate the basic principles of quantum mechanics using precise mathematical language. For more insights onto mathematical foundations of quantum mechanics we recommend [START_REF] Takhtadzhian | Quantum mechanics for mathematicians[END_REF]; Hall (2013). For a more physical exposition, see [START_REF] Cohen-Tannoudji | Quantum Mechanics[END_REF]. The following axioms constitute the basis of quantum mechanics.

(A1) With every quantum system is associated an infinite-dimensional separable Hilbert space H over C, with inner product •, • , the space of states. (A2) The set of observables A of a quantum system consists of all self-adjoint operators on H , i.e. A = {A :

H → H : Af 1 , f 2 = f 1 , Af 2 ∀(f 1 , f 2 ) ∈ H × H }. (A3)
The set of states S of a quantum system with a Hilbert space H consists of all trace class operators ρ with Tr ρ = 1. Pure states are projection operators onto one-dimensional subspaces of H . For ψ ∈ H with ψ = 1, the corresponding projection is denoted ρ ψ . All other states are called mixed states. (A4) Let P(R) denote the set of probability measures on R. A measurement is a mapping A × S (A, ρ) → µ A ∈ P(R), which to every observable A ∈ A and state ρ ∈ S assigns a probability measure µ A on R. For every Borel subset E ⊆ R, the quantity µ A (E) is the probability that for a quantum system in a state ρ the result of a measurement of the observable A lies in E.

To explicit the expression of the measure µ A in the last axiom, we need the following definition of a projection-valued measure (PVM).

Definition 1.12 -Let (X, X ) be a measured space, H a Hilbert space and B(H ) be the Banach space of bounded operators on H . A map µ : X → B(H ) is called a projectorvalued measure if the following properties are satisfied.

(1) For each E ∈ X , µ(E) is an orthogonal projection.

(2) µ(∅) = 0 and µ(X)

= I d . (3) If E 1 , E 2 , E 3 , . . . in X are disjoint, then for all f ∈ H we have µ(∪ ∞ j=1 E j )f = ∞ j=1 µ(E j )f
, where the convergence of the series is in the norm topology of H . ( 4) For all

E 1 , E 2 ∈ X , we have µ(E 1 ∩ E 2 ) = µ(E 1 )µ(E 2 ).
Suppose now, that f : (X, X ) → C is an unbounded measurable function, and we wish to define the integral of f with respect to the PVM µ : X → B(H ). First, note that every function ψ ∈ H describes a positive, real-valued measure µ ψ : X → [0, ∞] by setting µ ψ (E) := ψ, µ(E)ψ .

Then we have the following proposition, whose prove can be found in Hall (2013, section 10.1).

Proposition 1.3 -Suppose µ : X → B(H ) is a PVM and f : X → C is measurable (not necessarily bounded). Define a subspace D f of H by D f := ψ ∈ H : X |f (λ)| 2 dµ ψ (λ) < +∞ .
Then there exists a unique unbounded operator F : H → H with domain D f with the property that

ψ, F ψ = X f (λ) dµ ψ (λ)
for all ψ ∈ D f . The operator F is called the integral of f with respect to the PVM µ, which we also write as F = X f dµ.

With proposition 1.3 in mind, we can state Von Neumann's spectral theorem for unbounded self-adjoint operators, whose proof can be found in Hall (2013, section 10.1).

Theorem 1.5 -Suppose A is a self-adjoint operator on H , and let σ(A) denote the spectrum of A. Then there is a unique projection-valued measure P A on σ(A) with values in B(H ) such that

σ(A) λ dP A (λ) = A.
Going back to axiom (A4), the probability measure µ A is given by the Born-von Neumann formula

µ A (E) = Tr P A (E)ρ, ∀E ∈ B(R),
where P A is the unique PVM given by theorem 1.5. We now state the last axiom of quantum mechanics that is of interest for us (as a matter of completeness, other axioms deal with time evolution of quantum systems, which is of no interest for quantum homodyne tomography).

(A5) Observables A, B ∈ A can be measured simultaneously if and only if the corresponding PVM P A and P B satisfy P A (E 1 )P B (E 2 ) = P B (E 2 )P A (E 1 ) for all E 1 , E 2 ∈ B(R); in that case we say that observables A and B commute.

We finish this section with probably the most striking result of quantum mechanics, the so-called Heisenberg uncertainty relation, stated here for pure states only. Heisenberg's relation bound the precision at which two observables can be measured simultaneously; in particular, if the two observables are noncommuting, measuring one with precision is always to the detriment of the precision for the other.

Theorem 1.6 -Let A, B ∈ A and ρ ψ be a pure state with ψ ∈ dom(A) ∩ dom(B) and Aψ, Bψ ∈ dom(A)∩dom(B). Let σ 2 ψ (A), σ 2 ψ (B) denote respectively the variance of the observable A (respectively B) in the state ρ ψ . Then, σ 2 ψ (A)σ 2 ψ (B) ≥ i(AB -BA)ψ, ψ 2 /4. Finally, notice that the principles stated here cannot be verified directly. However, whenever the quantum mechanics is applicable, it is continuously verified that the experiments agree with the predicitions of the theory.

Statement of the problem

Here we make a brief statement of the problem of state estimation in Quantum Homodyne Tomography. For more details and motivations, we advice reading Artiles et al. (2005); Butucea et al. (2007).

We consider the quantum system of monochromatic light in a cavity, whose state is described by self-adjoint operator acting on the space of complex-valued square integrable functions on the real line L 2 (R).

The observables of interest for this quantum system are the electric and magnetic fields, whose corresponding self-adjoint operators on L 2 (R) are given by Q and, respectively P, with domains The derivative in the definitions of D(P) and P is understood in the distributional sense. The observables P, Q are noncommuting, hence from theorem 1.6 we cannot expect to have a joint distribution for the simultaneous measurement of P and Q on a system in state ρ. But, the Wigner transform of ρ, W ρ : R 2 → R, as defined below, is the closest object of a joint probability density function for the simultaneous measurement of P and Q [START_REF] Folland | Harmonic analysis in phase space[END_REF]. In particular, it is real valued and its marginals are probability density functions, with respect to Lebesgue measure on R. However, W ρ may contains patches of negative values, thus it cannot be a joint probability distribution in general. The Wigner transform of state ρ is defined by the property that the Fourier transform W ρ of W ρ with respect to both variables has the expression W ρ (x, y) := Tr(ρ exp(ixQ + iyP)).

D(Q) := {ψ ∈ L 2 (R) : x → xψ(x) ∈ L 2 (R)} and D(P) := {ψ ∈ L 2 (R) : x → ψ (x) ∈ L 2 (R
Although we cannot measure simultaneously the observables P and Q, it is possible to measure the quadrature observables, defined as X θ := Q cos θ + P sin θ for all θ ∈ [0, π]. We denote by X ρ θ the random variable whose distribution is the measurement of X θ on the quantum system in state ρ. Assuming that θ is drawn uniformly from [0, π], the joint probability density function (with respect to the Lebesgue measure on R × [0, π]) for (X ρ θ , θ) is given by the Radon transform of the Wigner distribution W ρ , that is

p ρ (x, θ) := 1 π R W ρ (x cos θ -ξ sin θ, x sin θ + ξ cos θ) dξ.
Given n quantum systems prepared in the same state ρ, the aim of Quantum Homodyne Tomography is to reconstruct ρ from n independent measurements of the quadrature observables X θ := Q cos θ + P sin θ, for some values of θ ∈ [0, π].

As mentioned in Butucea et al. (2007), the experimental setup for Quantum Homodyne Tomography is far from perfect, and we have to consider some noise addition in the statistical model to be realistic. In the same reference, authors justify that a good model is to correct the density p ψ by taking into account the efficiency η ∈ (0, 1) of the detector. The efficiency-corrected probability density is then the convolution

p η ρ (x, y) := 2 1 -η R p ρ (x, θ) exp - 2πη 1 -η (x -y) 2 dx. (1.6)
From a statistical perspective, we have n observations (Y 1 , θ 1 ), . . . , (Y n , θ n ) independently and identically distributed with density p η ρ (y, θ) (with respect to Lebesgue measure), where η ∈ (0, 1) is assumed to be known. Then the aim is to estimate W ρ , or better, to estimate ρ.

Frequentist results and minimax study

The problem of QHT is a statistical nonparametric ill-posed inverse problem that has been relatively well studied from a frequentist point of view in the last few years, and now quite well understood. We mention here only papers with theoretical analysis of the performance of their estimation procedure, which we summarize; many other physical papers references can be found therein.

We should classify frequentist methods in two categories, depending if they are based on estimating the state ρ, or estimating W ρ (although ρ → W ρ is one-to-one, methods based on estimating W ρ don't permit to do the reverse path from W ρ → ρ).

State estimation

The estimation of the state ρ from QHT measurements has been considered in the ideal situation (η = 1, no noise) by Artiles et al. (2005). To summarize their result, we need to introduce the Fock basis (ϕ j ) j∈N for L 2 (R),

ϕ j (x) := 1 √ π2 j j! H j (x)e -x 2 /2 ,
where H j are the Hermite polynomials. Artiles et al. (2005) used that the entries of the state ρ, expressed in the Fock basis, can be rewritten as

ρ j,k = 1 π +∞ -∞ π 0 p ρ (x, θ)f j,k (x)e -2πi(k-j)θ dθdx,
where f j,k are known functions called pattern functions, which exact expressions can be found in various papers of the quantum homodyne litterature. Then they proposed the estimator ρ (N,n) of ρ, such that the entries of ρ (N,n) expressed in the Fock basis are

ρ (N,n) j,k = n -1 n l=1 f j,k (Y l )e -2πi(k-j)θ l if 0 ≤ j, k ≤ N -1 0 if max(j, k) ≥ N.
The main result of Artiles et al. (2005) state that

E[Tr | ρ (N,n) -ρ| 2 ] → 0 as n → ∞, provided that N → ∞ with N = o(n 3/7 ).
The case where η = 1 is investigated in Aubry et al. (2008) under Frobenius-norm risk. They used a different estimation method depending on whether 0 < η ≤ 1/2 or 1/2 < η ≤ 1. We detail only the case 1/2 < η ≤ 1 here. They considered the estimator,

ρ N,η j,k = n -1 n l=1 f η j,k (Y l )e -2πi(k-j)θ l if 0 ≤ j + k ≤ N -1 0 otherwise. (1.7)
where f η j,k have Fourier transform f η j,k satisfying f η j,k (x) = f j,k (x)e -γx 2 , with γ := π(1η)/(2η) and f j,k the same pattern functions as previously. Then for the class of states

R(B, r, L) := ρ quantum state : |ρ j,k | ≤ L exp -B(j + k) r/2 ,
where the entries ρ j,k are expressed in the Fock basis, they proved that for a constant

C 0 > 0, lim sup n→∞ sup ρ∈R(B,r,L) ϕ -2 n E   j,k | ρ N,η j,k -ρ jk | 2   ≤ C 0 (1.8)
where

• ϕ 2 n = (log n) (12γ+B)/(12γ+3B) n -B/(4γ+B) , if r = 2 and N = log n 8γ+2B (1 + 2 3 log log n log n ). • ϕ 2 n = (log n) 2-r/3 e -2BN
r/2 , if 0 < r < 2 and N solution of 8γN + 2BN r/2 = log n. Clearly the estimator in equation (1.7) is not rate adaptive over R(B, r, L). Alquier et al. (2013) tackled the problem of adaptivity over R(B, r, L) using a soft-thresholded version of equation (1.7). For a prescribed tolerance level ∈ (0, 1), they estimated the state ρ by ρ η thres , where the entries of ρ η thres in the Fock basis are

ρ η thres (j, k) = ρ η j,k | ρ η j,k | (| ρ η j,k -t j,k ) + , t j,k = 2 f η j,k ∞ log 2N (N +1)
n , and ρ η j,k are computed as in equation (1.7). They found that there estimator is rateadaptive over all parameters L ≥ 1, B ≥ B 0 , r ≥ r 0 , where B 0 , r 0 can be chosen arbitrary small. They obtained up to additional power of log n factors, the same upper bound on the rates as in equation (1.8) Finally, let mention the the problem of goodness-of-fit testing for quantum state estimation is investigated in Méziani (2008).

Wigner density estimation

Regarding frequentist methods for estimating W ρ , the first result goes back to Guţă and Artiles (2007), where sharp minimax results for the pointwise risk are given over a class of ultra-smooth Wigner functions

A(β, r, L) := W ρ : | W ρ (z)| 2 e 2β z r dz ≤ L 2 ,
where W ρ is the Fourier transform of W ρ with respect to both variables. They proposed a kernel-type estimator based on the following band-limiter filter

K δ (x) := 1 2 δ -δ re 2πirx dr.
(1.9)

Their kernel estimator of W ρ based on the observations (Y 1 , θ n ), . . . , (Y n , θ n ) is W ρ (x, ω) := 1 n n l=1 K δn (x cos θ l + ω sin θ l -Y l ).
(1.10)

Then they found that the pointwise risk over A(β, 1, L) when

δ n = 2β/ log n satisfy the bound E[( W ρ (z) -W ρ (z)) 2 ] ≤ C(z)n -1 (log n) 3 (1 + o(1))
, where the constant C(z) > 0 is made precise in the original paper. More precisely, Guţă and Artiles (2007) provided a lower bound for the pointwise risk over a large subset of A(β, 1, L) that matches the previous upper bound, showing that the constant C(z) computed in their paper is sharp.

As far as we are not concerned with the case of ideal detection in this manuscript, we will not detail more this result.

The noisy framework has been considered in Butucea et al. (2007); authors proposed to modify the kernel of equation (1.9) to take into account the noise in the observations. They introduced instead3

K η h (x) = 1 2 1/h -1/h exp(-2πixt)|t| exp(-γt 2 ) dt,
where γ := π(1 -η)/(2η). Their estimator is very similar to equation (1.10),

W η ρ (x, ω) := 1 n n l=1 K η h (x cos θ l + ω sin θ l -Y l ). (1.11) They proved that if h is solution of 2β/h r + 2γ/h 2 = log n, then lim sup n→∞ sup z∈R 2 sup Wρ∈A(β,r,L) E[| W η h (z) -W ρ (z)| 2 ]ϕ -2 n ≤ C 0 ,
where

• C 0 = 1 and ϕ 2 n = √ Lh r-1 2βr exp -2β h r if 0 < r < 2. • C 0 > 0 and ϕ 2 n = n -β/(β+γ) if r = 2.
Furthermore, they also proved that these rates are minimax efficient for 0 < r < 2 and nearly minimax for r = 2, as they found the lower bounds lim inf

n→∞ inf Wρ sup Wρ∈A(β,r,L) E[| W ρ (z) -W ρ (z)| 2 ]ϕ -2 n ≥ 1, ∀z ∈ R 2 if 0 < r < 2 lim inf n→∞ inf Wρ sup Wρ∈A(β,r,L) E[| W ρ (z) -W ρ (z)| 2 ](n log n) β/(β+γ) > 0, ∀z ∈ R 2 if r = 2,
where the infinimum is taken over all possible estimators W ρ of W ρ . Finally, for 0 < r < 1, Butucea et al. (2007) established a bandwith h ad for which the estimator (1.11) is adaptive over all β > 0, 0 < r < 1 and L > 0.

In, Aubry et al. (2008), authors showed that the class R(B, r, L) defined in the previous section relates to A(β, r, L) in the following way. If 0 < r < 2 and β < 2 -r B, then R(B, r, L) ⊂ A(β, r, L ) for a suitable constant L > 0. For r = 2, the result is a little bit different, if 4β < B/(1 + √ B) 2 , then R(B, r, L) ⊂ A(β, r, L ) for a suitable L > 0. However, unlike Wigner functions in A(β, r, L), the Wigner functions for state in R(B, r, L) have very fast decay, which allowed Aubry et al. (2008) to derive an upper bound for the quadratic risk for estimating W ρ , ρ ∈ R(B, r, L), using a truncated version of the estimator (1.11). For a sequence s n → ∞, their estimator is

W η, * h := W η h (z)1( z ≤ s n ).
Then they proved that for a constant

C 0 > 0, lim sup n→∞ sup ρ∈R(B,r,L) E W η, * h -W ρ 2 2 ϕ -2 n ≤ C 0 ,
where Lounici et al. (2015) established the first sup-norm risk upper bound over A(β, r, L) with the estimator (1.11), as well as the first minimax lower bounds for both sup-norm and L 2 -norm risk over A(β, 2, L); showing in particular that (1.11) achieve optimal rates over A(β, 2, L) for both the sup-norm and L 2 -risk. Moreover, they also made the estimator (1.11) adaptive over all β, L > 0, r ∈ (0, 2], using a Lepski type procedure [START_REF] Lepski | Optimal pointwise adaptive methods in nonparametric estimation[END_REF]. As their paper is still under revision, we should not detail more their results. ψ 2 = 1, called the wave function, from noisy QHT measurements (Y 1 , θ 1 ), . . . , (Y n , θ n ). We will discuss in chapter 4 how the method can be extended to handle mixed-states estimation.

• ϕ 2 n = h 3r-10 exp(-2β(2h) -r ) if 0 < r < 2, β < B, s n = 1/h and h solution of the equation 2β(2h) -r + 2γh -2 = log n -(log log n) 2 . • ϕ 2 n = (log n) (16γ+3β)/(8γ+2β) n -β/(4γ+β) if r = 2, β = B/(1 + √ B) 2 , s n = 1/h and h 2 = 2 4γ+β log n + 1 4γ+β log log n. More recently,
The mixture of coherent states prior, as described in the next section, is motivated by [START_REF] Wolpert | A nonparametric bayesian approach to inverse problems[END_REF], where authors proposed to use mixture priors to address general inverse problems. Although we implement successfully mixtures of coherent states in practice, the posterior asymptotic, however, is still challenging today (this will be discussed more thoroughly at the end of chapter 4). Nevertheless, random Wilson series (section 1.4.4) may also be useful priors in practice, and an asymptotic study of this family of prior is made in chapter 4.

Coherent states mixtures

In quantum optics, a coherent state refers to a state of the quantized electromagnetic field that describes a classical kind of behavior.

Let T x f (y) := f (y -x), M ω f (y) = e 2πiωy f (y), denote the translation and modulation operators, respectively, and g a window function with g 2 = 1; most of time g is chosen as g(x) = 2 -1/4 exp(-πx 2 ). Mathematically speaking, coherent states are pure states ρ ψ , that is projection operators onto ψ ∈ L 2 (R), described by a wave-function

ψ belonging to ψ ∈ L 2 (R) : ψ = T x M ω g (x, ω) ∈ R 2 .
We suggest a mixture of coherent states as prior distribution on the wave function ψ. For a random measure Q on R 2 × [0, 2π], our model may be summarized by the following hierarchical representation. P η ψ denote the probability distribution having the density of equation (1.6), with ρ = ρ ψ .

(Y 1 , θ 1 ), . . . , (Y n , θ n ) i.i.d ∼ P η ψ , with ψ = ψ/ ψ 2 ψ(z) = R 2 ×[0,2π] e iφ T x M ω g(z) Q(dxdωdφ) Q ∼ Π.
Although this prior is appealing, in particular because all the states prepared in the lab have a very sparse representations in terms of coherent states, we are unable at this time to derive posterior contraction rates. However, simulation results given in chapter 4 show that these priors work well in practice.

Random Wilson series

For a, b > 0, g ∈ L 2 (R), the set of time-frequency shifts G(g; a, b) = {M n T m g : (n, m) ∈ aZ × bZ} is a Gabor frame if there exist positive constants A, B > 0, called the frame bounds, such that A f 2 2 ≤ m,n | f, T m M n g | 2 ≤ B f 2 2 , ∀f ∈ L 2 (R).
The frame operator

S g f := m,n f, T m M n g T m M n g is a bounded, positive and invert- ible mapping of L 2 (R) onto itself. Consequently, the canonical dual window h := S -1 g is also in L 2 (R). Moreover, from Gröchenig (2001, proposition 5.2.1) the canonical dual frame of G(g; a, b) is G(h; a, b) with frame bounds B -1 , A -1 , and for every f ∈ L 2 (R) f = m,n f, T m M n g T m M n h = m,n f, T m M n h T m M n g,
with unconditional convergence of the series in L 2 (R). Gabor expansions are not unique, which is appealing from a practical perspective, but appeared to be really annoying in establishing theoretical result. However, based on an idea of [START_REF] Wilson | Generalized wannier functions[END_REF], Daubechies et al. (1991) proposed a modification of Gabor systems for which the redundancy is removed, but good time-frequency localization properties are preserved. More precisely, they constructed a real-valued function ϕ such that for some a, b > 0,

|ϕ(x)| e -a|x| , | ϕ(ω)| e -b|ω| ,
and such that the ϕ lm , l ∈ N, m ∈ 1 2 Z defined by

ϕ lm (x) :=        ϕ(x -2m) if l = 0, √ 2ϕ(x -m) cos(2πlx) if l = 0 and 2m + l is even, √ 2ϕ(x -m) sin(2πlx) if l = 0 and 2m + l is odd,
constitute an orthonormal base for L 2 (R), namely the Wilson bases of exponential decay. Following Gröchenig (2001, section 8.5), we may rewrite ϕ lm in a convenient form for the sequel, emphasizing the relationship with coherent states and Gabor frames,

ϕ lm = c l T m (M l + (-1) 2m+l M -l )ϕ, (l, m) ∈ N × 1 2 Z,
where c 0 := 1/2 and c l := 1/ √ 2 for l ≥ 1. We then consider the following prior distribution Π on the wave function ψ. Let (ϕ lm ) be the previously defined orthonormal Wilson base with exponential decay. For any positive number Z, let Λ Z be the spherical array

Λ Z := (l, m) ∈ N × 1 2 Z : l 2 + m 2 < Z 2 .
Also define the simplex ∆ Z in the 2 metric as

∆ Z := p = (p lm ) (l,m)∈Λ Z : (l,m)∈Λ Z p 2 lm = 1, p lm ≥ 0 . Let P Z be a distribution over R + and draw Z ∼ P Z . Given Z, draw p from a distribution G(• | Z) over the simplex ∆ Z . Independently of p, draw ζ = (ζ lm ) (l,m)∈Λ Z from a distribution P ζ (• | Z) over [0, 2π] |Λ Z | and set ψ := (l,m)∈Λ Z
p lm e iζ lm ϕ lm .

Note that (ϕ lm ) is orthonormal, thus ψ 2 2 = (l,m)∈Λ Z p 2 lm = 1 almost-surely. This family of prior distribution may be seen as a discretized version of coherent states mixtures. However, unlike coherent states, Wilson bases have nice analytic properties that allow to challenge the posterior asymptotic in the model of QHT with noisy data.

Contribution to QHT

In addition to introduce the two Bayesian nonparametric approaches to the QHT problem described in the section 1.4.4, in chapter 4 we focus on evaluating posterior rates of convergence for random Wilson series distribution as prior over the set of pure statesequivalently the unit sphere of L 2 (R) -in the statistical model of equation (1.6).

To establish posterior rates of convergence, we need a suitable smoothness class. We propose to use closed balls in ultra-modulation spaces; which we describe here. To this aim, we need the following ingredients: the short-time Fourier transform, a class of windows and a class of weights. For a non-zero window function g ∈ L 2 (R), the short-time Fourier transform of a function f ∈ L 2 (R) with respect to the window g is given by

V g f (x, ω) := f, M ω T x g = R f (t)g(t -x)e -2πiωt dt.
We also need a class of analyzing windows g with sufficiently good time-frequency localization properties. We use the Gelfand-Shilov space

S 1 1 (R). A function f : R d → C belongs to the Gelfand-Shilov space S 1 1 (R d ) if f ∈ C ∞ (R d
) and there exist real constants h > 0 and k > 0 such that sup

x∈R d |f (x)e h x | < +∞, sup ω∈R d | f (ω)e k ω | < +∞.
Next, for β > 0, g ∈ S 1 1 (R), and r ∈ [0, 1), we consider the exponential weights on R 2 defined by x → exp(β x r ), and we introduce the class of wave-functions

C g (β, r, L) := ψ ∈ L 2 (R) : R 2 |V g ψ(z)| exp(β z r )dz ≤ L, ψ 2 = 1 .
It is shown in chapter 4 that C g (β, r, L) is independent of g in the sense that two functions g 1 , g 2 ∈ S 1 1 (R) lead essentially to the same classes of functions. Let Π denote the random Wilson series distribution over the unit sphere of L 2 (R). Assuming ψ 0 ∈ C g (β, r, L), our main theorem state sufficient conditions on

P Z , G(• | Z) and P ζ (• | Z) in the definition of Π to have M > 0 such that P η,n ψ 0 Π( ψ -ψ 0 2 ≥ M n | (Y 1 , θ 1 ), . . . , (Y n , θ n )) → 0,
where, for a constant a ≥ 0 depending only on the prior and γ

:= π(1 -η)/(2η), 2 n = (log n) 2a exp -β log n 2γ r/2
+ O( 1) .

(1.12)

It is not clear whether or not we can chose

P Z , G(• | Z) and P ζ (• | Z) to make these rates adaptive over (β, r, L) ∈ (0, ∞) × [0, 1) × (0, ∞)
. However, we construct in chapter 4 non adaptive examples where equation (1.12) is achieved with a = 2.

Introduction

Recently, interest in a Bayesian nonparametric approach to the sparse regression problem based on mixtures emerged from works of Abramovich et al. (2000), de Jonge and van Zanten (2010) and Wolpert et al. (2011). The idea is to model the regression function as

f (•) = X K(x; •)Q(dx), Q ∼ Π * , (2.1)
where K : X ×R d → R is a jointly measurable kernel function, and Π * a prior distribution on the space of signed measure over the measurable space X . Although the model (2.1) is popular in density estimation [START_REF] Escobar | Bayesian density estimation and inference using mixtures[END_REF]; [START_REF] Müller | Bayesian curve fitting using multivariate normal mixtures[END_REF]; Ghosal and van der Vaart (2007a); Shen et al. (2013); Canale and De Blasi (2017) and for modeling hazard rates in Bayesian nonparametric survival analysis [START_REF] Lo | On a class of bayesian nonparametric estimates: Ii. hazard rate estimates[END_REF]; [START_REF] Peccati | Linear and quadratic functionals of random hazard rates: An asymptotic analysis[END_REF]; De [START_REF] Blasi | Asymptotics for posterior hazards[END_REF]; [START_REF] Ishwaran | Computational methods for multiplicative intensity models using weighted gamma processes[END_REF]; [START_REF] Lijoi | A class of hazard rate mixtures for combining survival data from different experiments[END_REF], it seems that much less interest has been shown in regression.

Perhaps the little interest for mixture models in regression is due to the lack of variety in the choice of algorithms available, and in the insufficiency of theoretical posterior contraction results. To our knowledge, the sole algorithm existing for posterior simulations is to be found in Wolpert et al. (2011), when the mixing measure Q is a Lévy process.

On the other hand, The only contraction result available is to be found in de Jonge and van Zanten (2010) for a suitable semiparametric mixing measure.

Indeed, both designing an algorithm or establishing posterior contraction results heavily depends on the choice of K and Π * in equation (2.1); but above all also on the observation model we consider. This last point makes the study of mixtures in regression nasty to handle because of the diversity of observation models possible. In this article, we focus on the situation when Q is a symmetric Gamma process to propose both a new algorithm for posterior simulations and posterior contraction rates results.

In the first part of the paper, we propose a Gibbs sampler to get samples from the posterior distribution of symmetric Gamma process mixtures. The algorithm is sufficiently general to be used in all observation models for which the likelihood function is available. We begin with some preliminary theoretical result about approximating symmetric Gamma process mixtures, before stating the general algorithm. Finally, we make an empirical study of the algorithm, with comparison with the RJMCMC algorithm of Wolpert et al. (2011).

The second part of the paper is devoted to posterior contraction rates results. We consider the mean regression model with normal errors of unknown variance, and two types of mixture priors: location-scale and location-modulation. The latter has never been studied previously, mainly because it is irrelevant in density estimation models. However, we show here that it allows to get better rates of convergence than location-scale mixtures, and thus might be interesting to consider in regression.

Symmetric Gamma process mixtures

Let (Ω, E, P) be a probability space and (X , A) be a measurable space. We call a mapping

Q : Ω × A → R ∪{±∞} a signed random measure if ω → Q(ω, A) is a random variable for each A ∈ A and if A → Q(ω, A) is a signed measure for each ω ∈ Ω.
Symmetric Gamma random measures are infinitely divisible and independently scattered random measures (the terminology Lévy base is also used in Barndorff-Nielsen and Schmiegel (2004), and Lévy random measure in Wolpert et al. (2011)), that is, random measures with the property that for each disjoint A 1 , . . . , A k ∈ A, the random variables Q(A 1 ), . . . , Q(A k ) are independent with infinitely divisible distribution. More precisely, given α, η > 0 and F a probability measure on X , a symmetric Gamma random measure assigns to all measurable set A ∈ A random variables with distribution SGa(αF (A), η) (see section 2.A). Existence and uniqueness of symmetric Gamma random measures is stated in Rajput and Rosinski (1989).

In the sequel, we shall always denote by Π * the distribution of a symmetric Gamma random measure with parameters α, η and F , and we refer αF as the base distribution of Q ∼ Π * , and η as the scale parameter.

Location-scale mixtures

Given a measurable mother function g : R d → R, we define the location-scale kernel K A (x) := g(A -1 x), for all x ∈ R d and all A ∈ E, where E denote the set of all d×d positive definite real matrices. Then we consider symmetric Gamma location-scale mixtures of the type

f (x; ω) := E×R d K A (x -µ) Q(dAdµ; ω), ∀x ∈ R d , (2.2)
where

Q : B(E × R d ) × Ω → [-∞, ∞
] is a symmetric Gamma random measure with base measure αF on E × R d , and scale parameter η > 0. The precise meaning of the integral in equation (2.2) is made clear in Rajput and Rosinski (1989).

Location-modulation mixtures

As in the previous section, given a measurable mother function g :

R d → R, we define the location-modulation kernel K ξ,φ (x) := g(x) cos( d i=1 ξ i x i + φ), for all x ∈ R d , all ξ ∈ R d and all φ ∈ [0, π/2].
Then we consider symmetric Gamma location-modulation mixtures of the type

f (x; ω) := R d × R d ×[0,π/2] K ξ,φ (x -µ) Q(dξdµdφ; ω), ∀x ∈ R d , (2.3)
where

Q : B(R d × R d ×[0, π/2]) × Ω → [-∞, ∞] is a symmetric Gamma random measure with base measure αF on R d × R d ×[0, π/2],
and scale parameter η > 0.

Convergence of mixtures

Given a kernel K : X × R d → R and a symmetric Gamma random measure Q, it is not clear a priori whether or not the mixture y → K(x; y) Q(dx) converges or not, and in what sense. According to Rajput and Rosinski (1989) (see also Wolpert et al. (2011)), y → K(x; y) Q(dx) converges almost-surely at all y for which

R ×X (1 ∧ |uK(x; y)|)|u| -1 e -|u|η F (dx) < +∞.
Moreover, from the same references (or also in Kingman (1992)), if M is a complete normed space equipped with norm • , then

y → K(x; y) Q(dx) converges almost-surely in M if R ×X (1 ∧ |u| K(x; •) )|u| -1 e -|u|η F (dx) < +∞.
Since by definition F is a probability measure, we have for instance that the mixtures of equations (2.2) and (2.3) converges almost surely in L ∞ as soon as

K A ∞ < +∞ for F -almost every A ∈ E, or K ξ,φ ∞ < +∞ for F -almost every (ξ, φ) ∈ R d ×[0, π/2].

Simulating the posterior

In this section we propose a Gibbs sampler for exploration of the posterior distribution of a mixture of kernels by a symmetric Gamma random measure. The sampler is based on the series representation of the next theorem, inspired from a result about Dirichlet processes from Favaro et al. (2012), adapted to symmetric Gamma processes. In theorem 2.1, we consider M(X ) the space of signed Radon measures on the measurable space (X , A).

By the Riesz-Markov representation theorem (Rudin, 1974, Chapter 6), M(X ) can be identified as the dual space of C c (X ), the space continuous functions with compact support. That said, we endow M(X ) with the topology T v of weak-* convergence (sometimes referred as the topology of vague convergence), that is, a sequence {µ n ∈ M(X ) :

n ∈ N} converges to µ ∈ M(X ) with respect to the topology T v , if for all f ∈ C c (X ), X f (x) dµ n (x) → X f (x) dµ(x).
Dealing with prior distributions on M(X ), we shall equip M(X ) with a σ-algebra. Here it is always considered the Borel σ-algebra of M(X ) generated by T v . Before stating the main theorem of this section, we recall that a sequence of random variables {X i ∈ X : 1 ≤ i ≤ n} is a Pólya urn sequence with base distribution αF (•), where F is a probability distribution on (X , A) and α > 0, if for all measurable set A ∈ A,

P (X 1 ∈ A) = F (A), P (X k+1 ∈ A | X 1 , . . . , X k ) = F k (A)/F k (X ), k = 2, . . . , n -1,
where

F k := αF + k i=1 δ X i .
We are now in position to state the main theorem of this section, which proof is given in section 2.A.

Theorem 2.1 -Let X be a Polish space with Borel σ-algebra, p > 0 be integer, T ∼ Ga(α, η), independently, J 1 , . . . , J p i.i.d ∼ SGa(1, 1), and {X i ∈ X : 1 ≤ i ≤ p} a Pólya urn sequence with base distribution αF (•), independent of T and of the J i 's. Define the random measure, Q p := T /p p i=1 J i δ X i . Then Q p d → Q
, where Q is a symmetric Gamma random measure with base distribution αF (•) and scale parameter √ η.

Convergence of sequences of mixtures

In theorem 2.1, we proved weak convergence of the sequence of approximating measures (Q p ) p≥1 to the symmetric Gamma random measure, but it is not clear that mixtures of kernels by Q p also converge. The next proposition establish convergence in L q for general kernels, with 1 ≤ q < +∞, the proof is similar to the proof of Favaro et al. (2012, Theorem 2), thus we defer it into section 2.6.2. For any kernel K : X × R d → C, and any (signed) measure Q on (X , A), we write

f (Q) (y) := X K(x; y) Q(dx). Proposition 2.1 -If x → K(x; y)
is continuous for all x ∈ X , vanishes outside a compact set, and bounded by a Lebesgue integrable function h, then for any 1 ≤ q < +∞ we have lim p→∞ f (Qp) -f (Q) q = 0 almost-surely. Under supplementary assumptions on K, we can say a little-more about uniform convergence of the approximating sequence of mixtures. Assuming that y → K(x; y) is in L 1 for all x ∈ X , we denote by (x, u) → K(x; u) the L 1 Fourier transform on the second argument of (x, y) → K(x; y). Proposition 2.2 -Let y → K(x; y) be in L 1 for all x ∈ X and K satisfies the assumption of proposition 2.1. Then

lim p→∞ f (Qp) -f (Q) ∞ = 0 almost-surely.
Proof. We can assume without loss of generality that f (Qp) and f (Q) are defined on the same probability space (Ω, F, P). By duality, it is clear that f

(Qp) (•; ω) -f (Q) (•; ω) ∞ ≤ R d | f (Qp) (u, ω) -f (Q) (u; ω)| du,
where f denote the L 1 Fourier transform of f . Notice that by assumptions on K, f (Qp) and f (Q) are well-defined for almost all ω ∈ Ω (see section 2.2.3). Then by Fubini's theorem

f (Q p) (u; ω) = R d X K(x; y) Q(dx; ω) e -iuy dy = X R d K(x; y) e -iuy dy Q(dx; ω) = X K(x; u) Q(dx; ω),
and the conclusion follows from proposition 2.1.

General algorithm

From theorem 2.1, replacing Q by Q p for sufficiently large p, we propose a Pólya urn Gibbs sampler adapted from algorithm 8 in Neal (2000). In the sequel, we refer to Q p as the particle approximation of Q with p particles. Let Y = (Y i ) n i=1 be observations coming from a statistical model with likelihood function L(f |Y ), where f : R d → R is the regression function on which we put a symmetric Gamma mixture prior distribution. Let X = (X i ) p i=1 be a Pólya urn sequence, J := (J 1 , . . . , J p ) a sequence of i.i.d. SGa(1, 1) random variables, and T ∼ Ga(α, η) independent of (X i ) p i=1 and J. We introduce the clustering variables

C := (C 1 , . . . , C p ) such that C i = k if and only if X i = X k where X := X 1 , . . . stands for unique values of (X i ) p i=1 .
In the sequel, C -i stands for the vector obtained from removing the coordinate i to C, and the same definition holds for J mutatis mutandis. Given J, C, X, T and a measurable kernel K : X × R → R we construct f as

f (x) = T p p i=1 J i K(X i ; x).
We propose the following algorithm. At each iteration, successively sample from :

(1)

C i |C -i , Y, X, J, T , for 1 ≤ i ≤ p. Let n k,i = # 1≤l≤n l =i {C l = k}, κ (p) the number of distinct X k values and κ 0 a chosen natural, C i ind ∼ κ (p) k=1 n k,i L k,i (X, J, T |Y ) δ k (•) + α κ 0 κ 0 k=1 L k+κ (p) ,i (X, J, T |Y ) δ k+κ (p) (•),
where L k,i (X, J, T |Y ) stands for the likelihood under hypothesis that particle i is allocated to component k (note that the likelihood evaluation requires the knowledge of whole distribution F under any allocation hypothesis). ( 2) X|C, Y, J, T . Random Walk Metropolis Hastings on parameters.

(3) J i |J -i , K, Y, X, T , for 1 ≤ i ≤ p. Independent Metropolis Hastings with prior SGa(1, 1) taken as i.i.d. candidate distribution for J i . Note that for n → ∞, the posterior distribution of J i |J -i , C, Y, Z should be SGa(1, 1), then the number of particles p may be monitored using the acceptance ratio of the J i 's. (4) T |C, Y, X, J. Random Walk Metropolis Hastings on scale parameter.

Assessing the convergence of the Markov Chain

The previous algorithm produces a Markov Chain whose invariant distribution is (an approximation of) the posterior distribution of a symmetric Gamma process mixture. However, if the Markov Chain is initialized in a region of low posterior probability mass, we may over-sample this region. To avoid such over-sampling, we discard the first n 0 samples of the chain using Geweke's convergence diagnostic [START_REF] Geweke | Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments[END_REF].

More precisely, we monitor the convergence of the chain using the log-likelihood function. We start the algorithm with Markov Chain initialized at random from prior distribution. Then after n n 0 iterations we compute Geweke's Z-statistic for the log-likelihood using the whole chain; if the statistic is outside the 95% confidence interval we continue to apply the diagnostic after discarding 10%, 20%, 30% and 40% of the chain. If the Z-statistic is still outside 95% confidence interval, the chain is reported as failed to converge, and we restart the algorithm from a different initialization point.

Once we have discarded the first n 0 samples using Geweke's test, we run the chain sufficiently longer to get an Effective Sample Size (ESS) of at least 1000 samples, where we measure the ESS through the value of the log-likelihood at each iteration of the Markov Chain. A thinning of the chain is not required in general, however, we found in practice that a slight thinning improves the efficiency of the sampling.

In fig. 2.1, we draw some examples of temporal evolution of the log-likelihood on a simple univariate Gaussian mean regression problem. Here and after, we always choose step sizes in RWMH steps to achieve approximately 30% acceptance rates for each class of updates. Each subfigure represent 10 simulations with random starting point of the Markov Chain, distributed according to the prior distribution. We draw each subfigure varying the parameters liable to influence the mixing time of the chain, notably m and the number of particles. We observe that the speed at which the chain reach equilibrium is fast, especially when the number of particles is high. This last remark have to be balanced with the complexity in time of the algorithm which is O(mnp) for a naive implementation, and, depending on the nature of the likelihood, can be reduced to O(mp) or O(mp 2 ).

Examples of simulations

We now turn our attention to simulated examples to illustrate the performance of mixture models. First, we use mixtures as a prior distribution on the regression function in the univariate mean regression problem with normal errors. Of course, the interest for mixture comes when the statistical model is more involved. Hence, in a second time we present simulation results for the multivariate inverse problem of CT imaging.

Mean regression with normal errors

We present results of our algorithm on several standard test functions from the wavelet regression litterature (see [START_REF] Marron | Exact risk analysis of wavelet regression[END_REF], following the methodology from [START_REF] Antoniadis | Wavelet estimators in nonparametric regression: a comparative simulation study[END_REF] (i.e. Gaussian mean regression with fixed design and unknown variance). However, it should be noticed that mixtures are not a Bayesian new implementation of wavelet regression, and are much more general (see for instance the next section). For each test function, the noise variance is chosen so that the root signal-to-noise ratio is equal to 3 (a high noise level) and a simulation run was repeated 100 times with all simulation parameters constant, excepting the noise which was regenerated. We ran the algorithm for location-scale mixtures of Gaussians and Symmlet8, with normal N (0.5, 0.3) distribution as prior distribution on translations, and a mixture of Gamma distributions for scales (Ga(30, 0.06) and Ga(2, 0.04) with expectation 500 and 50 respectively). In addition of the core algorithm of section 2.3.2, we also added

• a Gibbs step estimation of the noise variance, with Inverse-Gamma prior distributon, • a Ga(2, 0.5) (with expectation 4) prior on α, with sampling of α done through a Gibbs update according to the method proposed in [START_REF] West | Hyperparameter estimation in Dirichlet process mixture models[END_REF], • a Dirichlet prior on the weights of the mixture of Ga(20, 0.2) and Ga(2, 0.1), with sampling of the mixture weights done through Gibbs sampling in a standard way, • a Ga(5, 10) (with expectation 0.5) prior on T , instead of normally Ga(α, η), which add more flexibility.

The choice of the mixture distribution as prior on scales may appear surprising, but we found in practice that using bimodal distribution on scales substantially improve performance of the algorithm, especially when there are few data available and/or high noise, because in general both large and small scales components are needed to estimate the regression function.

We ran the algorithm for n = 128 and n = 1024 data, and the performance is measured by its average root mean square error, defined as the average of the square root of the mean squared error n

-1 n i=1 | f (x i ) -f 0 (x i )| 2 ,
with f denoting the posterior mean and f 0 the true function. We ran on the same dataset the Translation-Invariant with hard thresholding algorithm (TI-H) and Symmlet8 wavelets (see [START_REF] Antoniadis | Wavelet estimators in nonparametric regression: a comparative simulation study[END_REF]), which is one of the best performing algorithm on this collection of test functions. We ran our algorithm with Symmlet8 kernels to make this comparison more relevant, since the choice of the kernel has major impact on the performance of the algorithm (see section 2.4.1 below).

Alternatives

In Wolpert et al. (2011), authors develop a reversible-jump MCMC scheme where the random measure is thresholded, i.e. small jumps are removed, yielding to a compound Poisson process approximation of the random measure, with almost-surely a finite number of jumps, allowing numerical computations. We also ran their algorithm with a thresholding level of = 0.05 (which seems to give the best performance), a Ga(15, 1) prior on η, and all other parameters being exactly the same as described in the previous section. We use the criteria of section 2.3.3 to stop the running of the chain.

Choosing the number of particles

It is not clear how to choose the number of particles in the algorithm. In theory, the higher is the better. In practice, however, we recommend choosing the number of particles according to the acceptance rate of particles weights move in step 3 of the algorithm. We found in practice that a level of acceptance between 20% and 30% is acceptable, as illustrated in fig. 2.2.

Simulation results

In tables 2.1 and 2.2 we summarize the results for location-scale mixtures of Gaussians and Symmlet8 produced by the algorithm of section 2.3.2 and by the RJMCMC algorithm of Wolpert et al. (2011), with the TI-H method as reference. We used p = 150 particles for both the datasets with n = 128 covariates and n = 1024 covariates, which is a nice compromise in terms of performance and computational cost. Regarding our algorithm and the RJMCMC algorithm, no particular effort was made to determine the value of the fixed parameters.

Obviously the Gibbs algorithm allow for sampling the full posterior distribution, pemitting estimation of posterior credible bands, as illustrated in figs.

Discussion

Obviously, the computation cost for our algorithm is high compared to TI-H, or any other classical wavelet thresholding method, even considering that it can intrinsically 2001), the choice of the kernel is crucial to the performance of estimators. The attractiveness of mixtures then comes because we are not restricted to location-scale or location-modulation kernels, and almost any function is acceptable as a kernel, which is not the case for most regression methods. Moreover, there is no requirements on how the data are spread, which makes the method interesting in inverse problems, such as in the next section.

Multivariate inverse problem example

Many medical imaging modalities, such as X-ray computed tomography imaging (CT), can be described mathematically as collecting data in a Radon transform domain. The process of inverting the Radon transform to form an image can be unstable when the data collected contain noise, so that the inversion needs to be regularized in some way.

Here we model the image of interest as a measurable function f : R 2 → R, and we propose to use a location-scale mixtures of Gaussians to regularize the inversion of the Radon transform. More precisely, the Radon transform

R f : R + ×[0, π] → R of f is such that R f (r, θ) = +∞ -∞ f (r cos θ-t sin θ, r sin θ+cos θ) dt. Then we consider the following model. Let n, m ≥ 1. Assuming that the image is supported on [-1, 1] 2 we let r 1 , . . . , r n equidistributed in [- √ 2, √ 2] and θ 1 , . . . , θ m equidistributed in [0, π]. Then, Y nm ∼ N (R f (r n , θ m ), σ 2 ) ∀n, m f ∼ Π,
where Π is a symmetric Gamma process location-scale mixture with base measure αF A × F µ on E ×R 2 , α > 0, and scale parameter η > 0. In the sequel, we use a normal distribution with mean zero and covariance matrix diag(τ, τ ) as distribution for F µ . Regarding F A , the choice is more delicate; we choose a prior distribution over the set of shearlet-type matrices of the form 1 s 0 1

a 0 0 √ a ,
where we set a N (1, σ 2 a ) distribution over the coefficient a and N (0, σ 2 s ) over the coefficient s. This type of prior distribution for F A is particularly convenient for capturing anisotropic features such as edges in images (Eas, 2009).

We ran our algorithm for n = 256 and m = 128 (32768 observations, a small amount), using the Shepp and Logan phantom as original image [START_REF] Shepp | The fourier reconstruction of a head section[END_REF]. The variance of the noise is σ 2 = 0.1, whereas the image take value between 0 and 2. Both the original image and the reconstruction are visible in fig. 2.5. Finally, we should mention that the choice of the Gaussian kernel for the mixture is convenient since it allows to compute the likelihood analytically. However, from a practical side, a full implementation of the algorithm with the intention of reconstructing CT images may benefit from using a different kernel.

Rates of convergence

In this section, we investigate posterior convergence rates in fixed design Gaussian regression for both symmetric Gamma location-scale mixtures and symmetric Gamma location-modulation mixtures.

Notations

In the sequel we use repeatedly the following notations.

• The conventional multi-index notation, for all α = (α 1 , . . . , α d ) ∈ N d and all z = (z 1 , . . . , z d ) ∈ R d we write |α| := α 1 +• • •+α d , α! := α 1 ! . . . α d !, and z α := z α 1 1 . . . z α d d .
Moreover, for all f : R d → R with continuous k-th order partial derivatives at x ∈ R d we write

D α f (x) := ∂ |α| f ∂z α 1 1 . . . ∂z α d n (x), |α| ≤ k.
• Let Ω be an open subset of R d and Ω be the closure of Ω. For any β > 0, we define C β (Ω), the Hölder space on Ω, as the set of all functions on Ω such that f

C β := max |α|≤k sup x∈Ω |D α f (x)| + max |α|=k sup x =y∈Ω |D α f (x) -D α f (y)|/|x -y| β-k is finite,
where k is the largest integer strictly smaller than β. • Given a signed measure µ on a measurable space (X , A), we let µ + and µ -denote respectively the positive and negative part of the Jordan decomposition of µ. Also, |µ| = µ + + µ -denote the total variation measure of µ. • Inequalities up to a generic constant are denoted by the symbols and .

The model

We consider the problem of a random response Y corresponding to a deterministic covariate vector x taking values in [-S, S] d for some S > 0. We aim at estimating the regression function

f : [-S, S] d → R such that f (x i ) = E Y i ,
based on independent observations of Y . More precisely, the nonparametric regression model we consider is the following,

Y i | i = f (x i ) + i , i = 1, . . . , n, 1 , . . . , n |σ 2 i.i.d
∼ N (0, σ 2 ), independently of (f, σ),

(f, σ) ∼ Π,
with Π the distribution on an abstract space Θ, given by σ ∼ P σ independently of f drawn from the distribution of a symmetric Gamma process mixture.

A general result

Let P θ,i denote the distribution of of Y i under the parameter θ = (f, σ), P n θ the joint distribution of (Y 1 , . . . , Y n ), P ∞ θ the distribution of the infinite sequence (Y 1 , . . . , Y ∞ ), and f 2 2,n := n -1 n i=1 |f (x i )| 2 . Let define the distance ρ n (θ 0 , θ 1 ) := f -f 0 2,n + | log σ 0 - log σ 1 |.
For the regression method based on Π, we say that its posterior convergence rate at θ 0 in the metric

ρ n is n if there is M < +∞ such that lim n→∞ Π ({θ ∈ Θ : ρ n (θ, θ 0 ) > M n }|Y 1 , . . . , Y n ) = 0 P ∞ θ 0 -a.s. (2.4)
Most of the approach to rates of convergence rely on idea coming from density mixtures models (Ghosal et al., 2000;Shen et al., 2013;Canale and De Blasi, 2017). Indeed, we prove that equation (2.4) hold by verifying a set of sufficient conditions established in theorem 2.2. For > 0 and any subset A of a metric space equipped with metric ρ, let N ( , A, ρ) denote the -covering number of A, i.e. the smallest number of balls of radius needed to cover A. Also, for all i = 1, . . . , n, define K i (θ 0 , θ) := (log dP θ 0 ,i /dP θ,i ) dP θ 0 ,i and V 2,i (θ 0 , θ) := (log dP θ 0 ,i /dP θ,i -K i (θ 0 , θ)) 2 dP θ 0 ,i , and let

K n (θ 0 , ) := θ : 1 n n i=1 K i (θ 0 , θ) ≤ 2 , 1 n n i=1 V 2,i (θ 0 , θ) ≤ 2 ,
be the Kullback-Leibler ball of size around θ 0 := (f 0 , σ 0 ). Theorem 2.2 is the analogue of theorem 5 in Ghosal and van der Vaart (2007b) for the Gaussian mean regression with fixed design ; the major difference reside on constructing suitable test functions, and extra cares have to taken regarding the fact that observations are not i.i.d. The proof of theorem 2.2 is given in section 2.7.

Theorem 2.2 -Let K := 3(32 ∨ 4σ 2 0 ) -1 , and n → 0 with n 2 n → ∞. Suppose that Θ n ⊂ Θ is such that Π(Θ c n ) e -3n 2 n for n large enough. Assume that Θ n ⊆ ∪ j Θ n,j is such that for some M > 0, lim n j N (M n , Θ n,j , ρ n ) Π(Θ n,j ) e -(KM 2 -2)n 2 n = 0, Π(K n (θ 0 , n )) e -n 2 n . Then Π(θ ∈ Θ : ρ n (θ 0 , θ) > 12M n |Y 1 , . . . , Y n ) → 0 in P n θ 0 -probability.

Supplementary assumptions

In order to derive rates of convergence (and only for this) we make supplementary assumptions on the choice of the mother function g and of the base measure αF .

Location-scale mixtures

We restrict our discussion to priors for which the following conditions are verified. We assume that

• g : R d → R is a non zero Schwartz function such that |g(x)| exp(-C 0 |x| τ d ) for some C 0 , τ > 0.
We assume that there is 0 ≤ γ < 1 such that sup |α|=k |D α g(0)| exp(γk log k) for all k large enough ; this last assumption is not obvious, it is for example met with γ = 1/2 if g is a multivariate Gaussian (see proposition 2.14 in appendix).

• αF := αF A × F µ , where F A is a probability measure on E s , the space of symmetric positive definite d × d reals matrices, and F µ a probability measure on [-2S, 2S] d . We also assume that there exist positive constants κ > 0,

κ * > d(d -1), a 1 , . . . , a 5 , b 1 , . . . , b 6 , C 1 , . . . , C 3 such that for any 0 < s 1 ≤ • • • ≤ s d , any z 0 ∈ [-2S, 2S] d , all
t ∈ (0, 1) and all x > 0 sufficiently large

F µ (z : |z -z 0 | ≤ t) ≥ b 1 t a 1 , (2.5) F A (A : λ d (A -1 ) ≥ x) ≤ b 2 exp(-C 2 x a 2 ), (2.6) F A (A : λ 1 (A -1 ) < 1/x) ≤ b 3 x -a 3 , (2.7) F A A : s j < λ j (A -1 ) < s j (1 + t), 1 ≤ j ≤ d ≥ b 4 s a 4 d t a 5 exp(-C 3 s κ/2 d ), (2.8) F A (A : λ 1 (A)/λ d (A) > x) ≤ b 6 x -κ * .
(2.9) Equations (2.6) to (2.8) are classical and are met for instance with κ = 2 if F A is the inverse-Wishart distribution (Shen et al., 2013, lemma 1). For a thorough discussion about equation (2.9) we refer to Canale and De Blasi (2017) and references therein. • P σ is a probability distribution on (0, ∞). We also assume that there are positive constants a 7 , a 8 , a 9 , b 7 , b 8 , C 8 , and b 9 eventually depending on σ 0 > 0, such that for all t ∈ (0, 1)

P σ (σ : σ > x) ≤ b 7 x -a 7 , (2.10) P σ (σ : σ ≤ 1/x) ≤ b 8 exp(-C 8 x a 8 ), (2.11) P σ (σ : σ 0 ≤ σ ≤ σ 0 (1 + t)) ≥ b 9 t a 9 .
(2.12)

Location-modulation mixtures

We restrict our discussion to priors for which the following conditions are verified. We assume that

• g : R d → R is a non zero Schwartz function such that g(x) ≥ 0 for all x ∈ R d and |g(x)| exp(-C 0 |x| τ d ) for some C 0 > 0 and τ > 1. We assume that there is a set E ⊆ [-π, π] d with strictly positive Lebesgue measure and a constant C > 0 such that g(x) ≥ C on E. We also assume that there is 0 ≤ γ < 1 such that sup |α|=k |D α g(0)| exp(γk log k) for all k large enough. As in the previous section, these assumptions are met for the multivariate Gaussian with E = [-π, π] d , γ = 1/2 and τ = 2 (see proposition 2.14 in appendix).

• αF := αF ξ × F µ × F φ , where F ξ is a probability measure on R d , F µ a probability measure on [-2S, 2S] d
, and F φ a probability measure on [0, π/2]. For all t ∈ (0, 1) and all z 0 ∈ [-2S, 2S] d we assume that F µ satisfies equation (2.5). We assume that there are positive constants a 10 , b 10 such that for all t ∈ (0, 1) and all

φ 0 ∈ [0, π/2] we have that F φ (φ : |φ -φ 0 | ≤ t) ≥ b 10 t a 10
. We also assume that there exist positive constants η > (d -1)/2, a 12 , a 13 , b 11 , b 12 such that for all t ∈ (0, 1), all ξ 0 ∈ R d and for all x > 0

F ξ (ξ : |ξ| d ≥ x) ≤ b 11 (1 + x) -2(η+1) (2.13) F ξ (ξ : |ξ -ξ 0 | d ≤ t) ≥ b 12 |ξ 0 | -a 12 d t a 13 .
(2.14)

• P σ satisfies the same assumptions of equations (2.10) to (2.12).

Results

Theorem 2.2 serves as a starting point for proving rates of contraction for symmetric Gamma process location-scale and location-modulation mixtures in the model of section 2.5.2. The proofs of the next theorems resemble to de Jonge and van Zanten ( 2010), but, they consider only a location mixture with locations taken on a lattice, allowing for a very specific construction of the sets Θ n . Here, we do not assume that locations are spread over a lattice, which makes the construction of Θ n more involved. Our construction is inspired from Shen et al. (2013) for Dirichlet processes mixtures, but adapted to symmetric Gamma processes (indeed, the same construction should work for many Lévy processes). Also, theorem 2.2 allows for partitioning Θ n onto slices Θ n,j , a step which is unnecessary for location mixtures (de Jonge and van Zanten, 2010; Shen et al., 2013), but yields to better rates and weaker assumptions on the prior when dealing with location-scale (Canale and De Blasi, 2017) and location-modulation mixtures.

Regarding the model of section 2.5.2, with deterministic covariates x 1 , . . . , x n arbitrary spread in [-S, S] d , we have the following theorem for location-scale mixtures. We notice that unlike de Jonge and van Zanten (2010), we do not assume that the covariates are spread on a strictly smaller set than [-S, S] d , i.e. the support of the covariates and the domain of the regression function are the same.

Theorem 2.3 -Let ζ = 1 ∨ 2/(τ -γτ ). Suppose that f 0 ∈ C β [-S, S] d for some S > 0.
Under the assumptions of section 2.5.4, the equation (2.4) holds for the location-scale prior with 2 n = n -2β/(2β+d+κ/2) (log n) 2βd(ζ-1)/(2β+d+κ/2) . Theorem 2.3 gives a rate of contraction analogous to the rates found in Canale and De Blasi (2017), that is to say, suboptimal with respect to the frequentist minimax rate of convergence. Indeed, if one use an Inverse-Wishart distribution for F A , then κ = 2; we can achieve κ = 1 with a distribution supported on diagonal matrices which assign square of inverse gamma random variables to non-null element of the matrix. Obviously, the choice of F A matters since it has a direct influence on the rates of contraction of the posterior. Also notice that the rates depends on κ/2, which is slightly better than the κ dependency found in Canale and De Blasi (2017). The reason is relatively artificial, since this follows from the fact that we put a prior on dilation matrices of the mixture, whereas they set a prior on square of dilation matrices (covariance matrices).

Location-modulation mixtures were never considered before, because they are not satisfactory for estimating a density. In comparison with location-scale mixtures, the major difference in proving contraction rates rely on approximating sufficiently well the true regression function. We use a new approximating scheme, based on standard of Fourier series analysis, yielding the following theorem.

Theorem 2.4 -Suppose that f 0 ∈ C β [-S, S] d for some S > 0. Under the assumptions of section 2.5.4, the equation (2.4) holds for the location-modulation prior with 2 n = n -2β/(2β+d) (log n) 2β(2d+1)/(2β+d) .

Although it was not surprising that location-scale mixtures yield suboptimal rates of convergence, we would have expected that location-modulation mixtures could be suboptimal too, which is not the case (up to a power of log n factor). Moreover, locationmodulation mixtures seem less stiff than location mixtures (Shen et al., 2013), hence they might be interesting to consider in regression.

Finally, it should be mentioned that all the rates here are adaptive with respect to β > 0; that is location-scale and location-modulation mixtures achieve these rates simultaneously for all β > 0.

2.6. Proofs of section 2.3

Preliminaries on convergence of signed random measures

It is well known for random (non-negative) measures that it is enough to show weak convergence of finite dimensional distributions on a semiring of bounded sets generating A to prove vague convergence of the distribution, see for instance Kallenberg (1983, Theorem 4.2) or Daley and Vere-Jones (2007, Theorem 11.1.VII). This fact remains true for random signed measures, but not in an obvious way. Indeed, it is well known that the vague topology is not metrizable on M(X ), even if X is Polish (for example, see Remark 1.2 in Del Barrio et al. ( 2007)), making the vague topology nasty to handle on M(X ). In particular, it is not as direct as in the case of non-negative measures to prove that the σ-algebra generated by the sets {{µ ∈ M(X ) :

µ(B) ∈ A} : A ∈ B(R), B ∈ R},
where R is a ring of bounded sets generating A, coincides with the Borel σ-algebra of M(X ), given the topology of vague convergence. However, once this last fact is proved, everything in the proof of Kallenberg (1983, Theorem 4.2) remains valid for signed random measures.

Surprisingly, there is not so much literature on vague convergence of signed random measures, and as our knowledge, the only reference available on this subject is [START_REF] Jacob | A representation of infinitely divisible signed random measures[END_REF]. We state here the result of interest for us, with only a sketch of the proof, as the details can be found in the original article.

(f ) ∈ A} : A ∈ B(R), f ∈ C c (X )}. Sketch of proof. First, we shall prove that S := σ{{µ ∈ M : µ(B) ∈ A} : A ∈ B(R), B ∈ R} = σ{{µ ∈ M : µ(f ) ∈ A} : A ∈ B(R), f ∈ C c (X )}.
Using the Hahn-Jordan decomposition of signed measures, this is a straightforward adaptation of Kallenberg (1983, Lemma 1.4).

Also, the argument of Kallenberg (1983, Lemma 4.1) for proving S ⊂ B(M) remains valid here, but the converse inclusion is not as direct. Let M + ⊂ M denote the cone of non-negative measures, and endow M + with the topology T + v of vague convergence (i.e.

µ n converges to µ if µ n (f ) → µ(f ) for any f ∈ C + c
) and corresponding Borel σ-algebra B(M + ). We denote S + the trace of S over M + . Hence, it suffices to prove that

(1) S + = B(M + ), (2) P : (M, S) → (M + × M + , S + × S + ), such that P (µ) := (µ + , µ -), is measurable, (3) R : (M + × M + , B(M + ) × B(M + )) → (M, B(M)), such that R(µ, ν) := µ -ν, is measurable.
These 3 conditions imply that R • P : (M, S) → (M, B(M)) is S/B(M)-measurable, and since R • P is just the identity mapping, this implies B(M) ⊂ S, as required.

Proofs

Proof of theorem 2.1. In the whole proof, we use the Pochhammer symbols x (n) and (x) n for respectively the nth power of the increasing factorial of x, and the nth power of the decreasing factorial of x. Once we took care of subtlety coming with section 2.6.1, the rest of the proof is identical to the proof of Proposition A.1 in Favaro et al. (2012), which we resume here for the sake of completeness. According to section 2.6.1 it is enough to check that

(Q p (A 1 ), . . . , Q p (A k )) d -→ (Q(A 1 ), . . . , Q(A k )), (2.15)
for any collection of disjoints bounded measurable sets A 1 , . . . , A k ∈ A, where Q is a symmetric Gamma random measure with parameters αF (•), η. Oviously, for any vector

(v 1 , . . . , v k ) ∈ R k the random variable v 1 Q(A 1 ) + • • • + v k Q(A k ) has symmetric Gamma distribution,
E [Q p (A 1 ) r 1 . . . Q p (A k ) r k ] -→ E [Q(A 1 ) r 1 . . . Q(A k ) r k ] (2.16)
holds for any disjoints bounded measurable sets A 1 , . . . , A k ∈ A and any positive integers r 1 , . . . , r k . From now, for all collection of measurable sets A 1 , . . . , A k ∈ A, we set A c := X \ ∪ k i=1 A i . We recall that if {X i ∈ X : i ≤ 1 ≤ p} is a Pólya urn sequence with base distribution αF (•), and A 1 , . . . , A k ∈ A are disjoints, then P (#{i :

X i ∈ A 1 } = j 1 , . . . , #{i : X i ∈ A k } = j k ) = p j 1 . . . j k (αF (A 1 )) (j 1 ) . . . (αF (A k )) (j k ) (αF (A c )) (p-k i=1 j i ) (p -k i=1 j i )! α (p)
, where (j 1 , . . . , j k ) ∈ E k,p , with E k,p := {(j 1 , . . . , j k ) ∈ {0, . . . , p} k : k i=1 j i ≤ p}. It is straightforward to show that both the lhs and the rhs of equation (2.16) are null whenever one of the r i 's is odd. Therefore we shall only consider equation (2.16) for even exponents. We deduce from proposition 2.10 that for any disjoints bounded measurable sets A 1 , . . . , A k ∈ A and any positive integers r 1 , . . . , r k ,

E Q p (A 1 ) 2r 1 . . . Q p (A k ) 2r k = α (r 1 +•••+r k ) k i=1 (2r i )!/r i ! ( √ η) 2r i p r i × (j 1 ,...,j k )∈E k,p p j 1 . . . j k (αF (A 1 )) (j 1 ) . . . (αF (A k )) (j k ) (αF (A c )) (p-k i=1 j i ) (p -k i=1 j i )! α (p) × (j 1 ) (r 1 ) . . . (j k ) (r k ) .
Introducing s(•, •) and S(•, •) are the Stirling numbers of the first and second kind, we can mimic Favaro et al. (2012, Appendix A.1) to find that

E Q p (A 1 ) 2r 1 . . . Q p (A k ) 2r k = α (r 1 +•••+r k ) k i=1 (2r i )!/r i ! ( √ η) 2r i p r i × r 1 m 1 =0 |s(r 1 , m 1 )| m 1 s 1 =0 S(m 1 , s 1 ) • • • r k m k =0 |s(r k , m k )| m k s k =0 S(m k , s k ) × (αF (A 1 )) (s 1 ) . . . (αF (A k )) s k α (s 1 +•••+s k ) (p) s 1 +•••+s k .
Therefore, we conclude that,

lim p→∞ E Q p (A 1 ) 2r 1 . . . Q p (A k ) 2r k = k i=1 (2r i )! r i ! (αF (A i )) (r i ) ( √ η) 2r i .
Proof of proposition 2.1. We can assume that all the Q p and Q are defined on the same probability space (Ω, F, P). The proof is an adaption of Favaro et al. (2012, Theorem 2).

We just have to take care that here, we proved Q p → Q vaguely in theorem 2.1, which does not necessarily imply that f (Qp) (x) → f (Q) (x) pointwise. But by assumption, x → K(x; y) is continuous and vanishes outside a compact set, and it is easily seen that the sequence of total mass |Q|(•; ω) is almost-surely bounded, then by Bauer (2001, Theorem 30.6) (which remains valid for signed measures), we have

f (Qp) (x) → f (Q) (x) pointwise, almost-surely.
The end of the proof is identical to Favaro et al. (2012, Theorem 2) for convergence in L 1 , and extension to L q with 1 ≤ q < +∞ is straightforward.

Proofs of section 2.5.3

Proof of theorem 2.2. The proof is similar to Ghosal and van der Vaart (2007b, theorem 5). The event Ghosal and van der Vaart (2007a) and assumptions on Π. Therefore,

A n that n i=1 dP θ,i dP θ 0 ,i (Y i ) dΠ(θ) ≥ e -2n 2 n /2 satisfies P n θ 0 (A c n ) → 0 by Lemma 10 in
P n θ 0 Π(Θ c n |Y 1 , . . . , Y n ) ≤ P n θ 0 [Π(Θ c n |Y 1 , . . . , Y n ) 1 An ] + P n θ 0 (A c n ) ≤ e 2n 2 n P n θ 0 Θ c n n i=1 dP θ,i dP θ 0 ,i (Y i ) dΠ(θ) + P n θ 0 (A c n ) ≤ e 2n 2 n Π(Θ c n ) + P n θ 0 (A c n ) → 0,
where the last lines follows by Fubini's theorem. For 0 < α j ≤ 1, and n large enough, the lemma 2.2 states the existence of tests functions ψ n,j such that

P n θ 0 ψ n,j ≤ 2α j N (M n , Θ n,j , ρ n ) e -KM 2 n 2 n , P n θ (1 -ψ n,j ) ≤ α -1 j e -KM 2 n 2 n , for all θ ∈ Θ n,j with ρ n (θ, θ 0 ) > 12M n . Letting U := {θ ∈ Θ : ρ n (θ 0 , θ) > 12M n }, P n θ 0 [Π(U ∩ Θ n,j |Y 1 , . . . , Y n ) 1 An ] ≤ P n θ 0 ψ n,j + P n θ 0 (1 -ψ n,j ) U ∩Θ n,j n i=1 dP θ,i dP θ 0 ,i (Y i ) dΠ(θ) e 2n 2 n ≤ P n θ 0 ψ n,j + sup U ∩Θ n,j P n θ (1 -ψ n,j )Π(Θ n,j ) e 2n 2 n ≤ 2α j N (M n , Θ n,j , ρ n ) e -KM 2 n 2 n +α -1 j Π(Θ n,j ) e -(KM 2 -2)n 2 n ,
where we used Fubini's theorem again. Put α j = Π(Θ n,j )/N (M n , Θ n,j , ρ n ) (notice that α j ≤ 1) and sum over j to obtain the result in view of the last equation.

Existence of tests

Here we construct the test functions required in the proof of theorem 2.2. We proceed in two steps. First, we construct tests for testing the hypothesis that θ = θ 0 against θ belongs to a ball of radius /12 centered at θ 1 with ρ n (θ 0 , θ 1 ) > ; then in lemma 2.2 we construct the tests used in the proof of theorem 2.2.

Let θ 0 = (f 0 , σ 0 ), θ 1 = (f 1 , σ 1 ), θ 10 = (f 1 , σ 0 ), δ = 2 + (108/n) log(1/α),

and define,

A n := y ∈ R n : n i=1 log dP θ 0 ,i dP θ 10 ,i (y i ) < - n 2 96σ 2 0 + 2 log α , B c n := y ∈ R n : n(1 -δ/3) ≤ n i=1 y i -f 0 (x i ) σ 0 2 ≤ n(1 + δ/3) .
Then we construct the sequence (φ n ) n≥0 as

φ n (Y 1 , . . . , Y n ) := 1 An (Y 1 , . . . , Y n ) + 1 Bn (Y 1 , . . . , Y n ) -1 An (Y 1 , . . . , Y n ) 1 Bn (Y 1 , . . . , Y n ). Proposition 2.3 -Let K = 3(32 ∨ 4σ 2 0 ) -1 .
The tests φ n defined above satisfy P n θ 0 φ n ≤ e -Kn 2 /144 and sup θ∈Θ : ρn(θ,θ 1 )< /12 P n θ (1 -φ n ) ≤ e -Kn 2 /144 for all θ 1 ∈ Θ such that ρ n (θ 0 , θ 1 ) > and all 0 < ≤ 1.

Proof. Type I error of φ n . It is clear that

P n θ 0 φ n ≤ P n θ 0 (A n ) + P n θ 0 (B n ).
Moreover, by proposition 4 in [START_REF] Birgé | Model selection via testing: an alternative to (penalized) maximum likelihood estimators[END_REF], we have

P n θ 0 (A n ) ≤ α e -n 2 /(192σ 2 0 )
, and regarding the proof of lemma 7 in [START_REF] Choi | On posterior consistency in nonparametric regression problems[END_REF], the bound P n θ 0 (B n ) ≤ 2 e -nδ 2 /108 = 2α e -n 2 /108 holds for n sufficiently large.

Type II error of φ n . Let θ = (f, σ) be such that ρ n (θ, θ 1 ) ≤ /12. Clearly,

P n θ (1-φ n ) = P n θ (1 -1 An )(1 -1 Bn ) ≤ P n θ (A c n ) ∧ P n θ (B c n ). We should consider two situations, either | log σ 0 -log σ 1 | ≤ /2, or | log σ 0 -log σ 1 | > /2. • If | log σ 0 -log σ 1 | ≤ /2, then ρ n (θ 0 , θ 1 ) > implies f 0 -f 1 2,n > /2,
and for all θ with ρ n (θ, θ 1 ) ≤ /12, it is clear that f -f 1 2,n ≤ /12. It follows from proposition 4 in [START_REF] Birgé | Model selection via testing: an alternative to (penalized) maximum likelihood estimators[END_REF] that

P n θ (A c n ) ≤ exp - n f 0 -f 1 2 2,n -n 2 /8 + 24σ 2 0 log α 24σ 2 0 ≤ 1 α exp - n 2 64σ 2 0 . • If | log σ 0 -log σ 1 | > /2, then ρ n (θ, θ 1 ) ≤ /12 implies | log σ -log σ 0 | > 5 /12.
We should again subdivise this case, considering either σ/σ 0 ≥ 1 or not. For both cases we mimick and adapt the proof of lemma 7 in [START_REF] Choi | On posterior consistency in nonparametric regression problems[END_REF].

-

If σ/σ 0 ≥ 1, because | log σ -log σ 0 | > /3 we have σ > σ 0 e /3
, and thus σ > (1 + /3)σ 0 for any > 0. Let W ∼ χ 2 n and let W have a noncentral χ 2 distribution with n degrees of freedom and noncentrality parameter

n i=1 (f (x i ) -f 0 (x i )) 2 . Then, P n θ (B c n ) ≤ P n θ n i=1 Y i -f 0 (x i ) σ 0 2 ≤ n 1 + δ 3 = P W ≤ n σ 2 0 σ 2 1 + δ 3 ≤ P W ≤ n σ 2 0 σ 2 1 + δ 3 .
But whenever 0 < α ≤ 2, we have

σ 2 0 σ 2 1 + δ 3 ≤ 1 + δ/3 (1 + /3) 2 ≤ 1 1 + /3 + (108/n) log(1/α) 6 (1 + /3) 2 .
Therefore, by Markov's inequality we get for all t < 1/2

P n θ (B c n ) ≤ exp -t 108 log(1/α) 6 (1 + /3) 2 exp - nt 1 + /3 (1 -2t) -n/2 .
Choosing t = -/18 leads to

P n θ (B c n ) ≤ exp log(1/α) (1 + /3) 2 exp n 2 /9 1 + /3 -log(1 + /9) ≤ 1 α exp - 7n 2 648 ≤ 1 α exp - n 2 93 ,
because we have 0 < ≤ 1. This concludes the proof when σ/σ 0 ≥ 1. -On the other direction, σ/σ 0 < 1 and | log σ -log σ 0 | > 5 /12 imply that σ < (1 -/3)σ 0 for any 0 < ≤ 1. Using the same strategy as in the previous item it is possible to show that the bound

P n θ (B c n ) ≤ (1/α) e -n 2 /1536 holds. Lemma 2.2 -Let Θ n ⊂ Θ and K := 3(32 ∨ 4σ 2 0 ) -1 .
Then for any 0 < α ≤ 1 there exists a collection of tests functions (ψ n ) n≥1 such that for any 0 < ≤ 1/12 and any n ≥ 1

P n θ 0 ψ n ≤ 2αN ( , Θ n , ρ n ) e -Kn 2 , sup θ∈Θn : ρn(θ,θ 0 )>12 P n θ (1 -ψ n ) ≤ α -1 e -Kn 2 .
Proof. Let N ≡ N ( /12, Θ n , ρ n ) denote the number of balls of radius /12 needed to cover Θ n . Let (B 1 , . . . , B N ) denote the corresponding covering and (ζ 1 , . . . , ζ N ) denote the centers of (B 1 , . . . , B N ). Now let J be the index set of balls B j with ρ n (θ 0 , ζ j ) > . Using proposition 2.3 for 0 < ≤ 1 and for any ball B j with j ∈ J, we can build a test function φ n,j satisfying P n θ 0 φ n,j ≤ 2α e -Kn 2 /144 , sup

θ∈B j P n θ (1 -φ n,j ) ≤ α -1 e -Kn 2 /144 , Let ψ n := max j∈J φ n,j . Then P n θ 0 ψ n ≤ j∈J P n θ 0 φ n,j ≤ 2αN ( /12, Θ n , ρ n ) e -Kn 2 /144 and also P n θ (1 -ψ n ) ≤ min j∈J sup θ ∈B j P n θ (1 -φ n,j ) ≤ α -1 e -Kn 2 /144 for any θ ∈ Θ n with ρ n (θ, θ 0 ) > .

Proof of theorem 2.3

We prove theorem 2.3 by verifying the set of sufficient conditions established in theorem 2.2.

Sieve construction

For constants H, M > 0 to be determined later, we define the sets

D n := A ∈ E s : n -1/a 2 ≤ λ i (A) ≤ n -1/a 2 (1 + M n /n) n 2 , i = 1, . . . , d , Θ n :=          (f, σ) : n -2/a 8 < σ 2 ≤ n -2/a 8 (1 + M n ) n , f (x) = K A (x -µ)Q(dAdµ), Q = ∞ i=1 u i δ A i ,µ i , supp Q = E s × [-2S, 2S] d , ∞ i=1 |u i | ≤ n, #{i : |u i | > n -1 , A i ∈ D n } ≤ Hn 2 n / log n, ∞ i=1 |u i | 1{A i / ∈ D n } ≤ M n , ∞ i=1 |u i | 1{|u i | ≤ n -1 } ≤ M n          .
In the sequel, we assume without loss of generality that the jumps of Q in the definition of Θ n are ordered so that there is no jump with |u i | > n -1 and A i ∈ D n when i > Hn 2 n / log n. Moreover, we consider the following partition of Θ n . Let H n the largest integer smaller than Hn 2 n / log n. Then for any j = (j 1 , . . . , j Hn ) ∈ {1, 2, . . .} Hn , inspired by Canale and De Blasi (2017, theorem 2) we define the slices

Θ n,j := (f, σ) ∈ Θ n : n 2 j i -1 < λ 1 (A i )/λ d (A i ) ≤ n 2 j i ∀i ≤ H n . Lemma 2.3 -Assume that there is 0 < γ 1 < 1 such that 2 n ≥ n -γ 1 for all n large enough. Then for H = 6(1 -γ 1 ) -1 it holds Π(Θ c n ) exp(-3n 2 n ) as n → ∞.
Proof. From the definition of Θ n , it is clear that

Π(Θ\Θ n ) ≤ Π #{i : |u i | > n -1 } > Hn 2 n / log n + Π ( ∞ i=1 |u i | > n) + Π ∞ i=1 |u i | 1{|u i | ≤ n -1 } > M n + Π ( ∞ i=1 |u i | 1{(A i , µ i ) / ∈ D n } > M n ) + P σ (σ 2 ≤ n -2/a 8 ) + P σ (σ 2 > n -2/a 8 (1 + M n ) n ).
(2.17)

The bounds on the two last terms are obvious in view of equations (2.10) and (2.11).

By the superposition theorem (Kingman, 1992, section 2), for any measurable set

A ⊆ E × R d we have Q(A) := Q 1 (A) + Q 2 (A)
where Q 1 and Q 2 are independent signed random measures with total variation having Laplace transforms (for all measurable A ⊆ E × R d and all t ∈ R for which the integrals in the expression converge)

E e t|Q 1 |(A) = exp 2αF (A) ∞ n -1
(e tx -1)x -1 e -ηx dx ,

(2.18)

E e t|Q 2 |(A) = exp 2αF (A) n -1
0 (e tx -1)x -1 e -ηx dx .

(2.19)

The random measures Q 1 and Q 2 are almost-surely purely atomic, the magnitudes of the jumps of Q 1 are all ≥ n -1 , whereas Q 2 has jumps magnitudes all < n -1 (almost-surely). Also, the number of jumps of Q 1 is distributed according to a Poisson law with intensity αE 1 (n -1 /η), where E 1 is the exponential integral E 1 function. Recalling that E 1 (x) γ + log(1/x) for x small, it follows α(γ + log η) ≤ αE 1 (n -1 /η) ≤ 2α log n Hn 2 n / log n when n is large. Then using Chernoff's bound on Poisson law, we get Π #{i :

|u i | > n -1 } > Hn 2 n / log n ≤ e -αE 1 (n -1 /η) (e αE 1 (n -1 /η) Hn 2 n / log n (Hn 2 n / log n) Hn 2 n / log n ≤ (ηe γ ) α exp - Hn 2 n log n log Hn 2 n log n -log(2eα log n) . But, log Hn 2 n log n -log(2eα log n) ≥ (1 -γ 1 ) log n -2 log log n + log H 2eα ,
which is in turn greater than (1/2)(1 -γ 1 ) log n when n becomes large. This gives the proof for the first term of the rhs of equation (2.17).

Regarding the second term of the rhs of equation (2.17), it suffices to remark that the random variable n i=1 |u i | has Gamma distribution with parameters (2α, η). Then the upper bound on Π( n i=1 |u i | > n) follows from Markov's inequality. With the same argument, we have that the random variable ∞

i=1 |u i | 1{|u i | ≤ n -1 } is equal in distri- bution to |Q 2 |(E × R d )
, thus the bound for the fourth term of the rhs of equation (2.17) follows from Markov's inequality and equation (2.19), because

Π e 3n n|Q2| > e 3n 2 n ≤ e -3n 2 n exp 2α n -1 0 (e n nx -1)x -1 e -ηx dx e -3n 2 n .
The fifth term of the rhs of equation (2.17) is bounded using Chebychev's inequality. Indeed, with the same argument as before, the random variable

X := n i=1 |u i | 1{A i / ∈ D n } has Gamma distribution with parameters (2αF A (D c n ), η). Hence for n sufficiently large we have EX = 2αF A (D c n )/η ≤ n /2, and Π(X > n ) ≤ Π(X -EX > n /2) ≤ 8αF A (D c n ) η 2 2 n .
Then the result follows from equations (2.6) and (2.7).

Lemma 2.4 -Let n → 0 with n 2 n → ∞ and K = 3(32 ∨ 4σ 2 0 ). Then there exists M > 0 such that it holds j N (M n , Θ n,j , ρ n ) Π(Θ n,j ) e -(KM 2 -2)n 2 n → 0.

Proof. Define the random measures Q 1 and Q 2 as in the proof of lemma 2.3. Then using the Poisson construction of Q 1 (see for instance Wolpert et al. (2011, section 2.3.1)), it follows from equation (2.9) that for any j ∈ {1, 2, . . .} Hn

Π(Θ n,j ) ≤ i≤Hn F A (A : λ 1 (A)/λ d (A) ≥ n 2 j i -1 ) ≤ b Hn 6 i≤Hn n -κ * 2 j i -1 .
Moreover, using proposition 2.4 we can find a constant

C > 0 independent of M such that N (M n , Θ n,j , ρ n ) ≤ e -2CHn 2 n n d(d-1
)/2 i≤Hn 2 j i when n is large. Therefore,

N (M n , Θ n,j , ρ n ) Π(Θ n,j ) ≤ exp Hn 2 n C + log b 6 2 log n i≤Hn n 1 2 [d(d-1)-κ * ]2 j i -1 .
For n large enough we have log b 6 ≤ 2C log n ; then provided κ * > d(d -1), we can sum over j ∈ {1, 2, . . .} Hn the last expression to get

j N (M n , Θ n,j , ρ n ) Π(Θ n,j ) ≤ exp 2CHn 2 n k≥1 n 1 2 [d(d-1)-κ * ]2 k-1 Hn ≤ exp H(2C + κ * /2)n 2 n . Now choose M > 0 satisfying KM 2 > 2 + H(2C + κ * /2
) to obtain the conclusion of the lemma.

Proposition 2.4 -For n large enough there is a constant C > 0 independent of M such that for any sequence n → 0 with n 2 n → ∞, the following holds for any j ∈ {1, 2, . . .} Hn .

log

N (M n , Θ n,j , ρ n ) ≤ CHn 2 n + d(d -1) 2 log n i≤Hn 2 j i .
Proof. The proof is based on arguments from Shen et al. (2013), it uses the fact that the covering number

N (M n , Θ n , ρ n ) is the minimal cardinality of an M n -net over Θ n in the distance ρ n . Let δ n := M n n -(1+1/a 2 ) , R n be a δ n -net of [-2S, 2S] d , ∆ n be a M n -net of {(u 1 , . . . , u Hn ) ∈ R Hn : Hn i=1 |u i | ≤ n} in the 1 -distance, and S n := {σ > 0 : σ 2 = n -2/a 8 (1 + M n ) k , k ∈ N, k ≤ n}. Also, for any k ≥ 1 let O k be a n -(2 k +1) M n -
D n,k := A ∈ D n : A = P ΛP , P ∈ O k , Λ = diag(λ 1 , . . . , λ d ), λ j = n -1/a 2 (1 + M n /n) k , k ∈ N, k ≤ n 2 , j = 1, . . . , d . Pick (f, σ) ∈ Θ n,j with f (x) = ∞ i=1 u i K A i (x -µ i ). Clearly we can find u ∈ ∆ such that i≤Hn |u i -u i | ≤ M n , µ ∈ R Hn n such that |µ i -µ i | d ≤ δ n for all i = 1, . . . , H n , and σ ∈ S n such that | log σ -log σ| ≤ M n .
We also claim that we can find

A i ∈ D n,j i such that I -A -1 i A -1 i ≤ 3dM n /n for all i ≤ H n .
We defer the proof of the claim to later. Let f (x) = i≤Hn u i K A i (x -µ i ) denote the function built from the parameters chosen as above ; it follows

f -f 2,n ≤ i>Hn |u i | + i≤Hn |u i -u i | + i≤Hn |u i | K A i (• -µ i ) -K A i (• -µ i ) 2,n ≤ 2M n + C i≤Hn |u i | I -A -1 i A i + C i≤Hn |u i | A -1 i |µ i -µ i | d ≤ M (2 + C + 3C d) n ,
where the two last inequalities hold by proposition 2.12 for a constant C > 0 depending only on g, and because

A -1 i ≤ n 1/a 2 for all i ≤ H n . Thus a (2 + C + 3C d)M n - net of Θ n,j in the distance ρ n can be constructed with ( f , σ) as above. Recall that # R n ≤ (4S/δ n ) d , # ∆ n ≤ (n/(M n )) Hn , # S n = n and # O k ≤ (n -(2 k +1) M n ) -d(d-1)/2 . It turns out that # D n,k ≤ n 2d × # O k . Then the total number of ( f , σ) is bounded by a multiple constant of n × 4S δ n Hn × n M n Hn i≤Hn   n 2d × n 2 j i +1 M n d(d-1)/2   .
Finally, H n | log M | H n log n when n is large proving that the constant C > 0 can be chosen independent of M , and the constant factor 2 + C + 3C d can be absorbed into the bound.

It remains to prove that for any

A ∈ D n with λ 1 (A)/λ d (A) ≤ n 2 k we can find A ∈ D n,k such that I -A -1 A ≤ 3dM n /n.
Let A =: P ΛP denote the spectral decomposition of A (recall that A is symmetric). Clearly, we can find a matrix A := P Λ P in D n,k with P -P ≤ n -(2 k +1) M n and 1 ≤ λ j (Λ)/λ j ( Λ) ≤ 1 + M n /n for all j = 1, . . . , d.

Let A := P Λ P and remark that

I -A -1 A ≤ I -A -1 A + A -1 A I -A -1 A ≤ I -A -1 A + I -A -1 A (1 + I -A -1 A ).
(2.20)

Let B := P P -I, so that B max ≤ B ≤ P P -

I ≤ P -P ≤ n -(2 k +1) M n , and
I -A -1 A = P (B -Λ -1 BΛ) P . It follows, I -A -1 A ≤ B -Λ -1 BΛ ≤ d B max λ 1 (Λ) λ d (Λ) ≤ dM n /n, because the entries of B -Λ -1 BΛ are equal to B ij (1 -Λ j /Λ i ) and • ≤ d • max . Moreover, I -A -1 A = P (I -Λ -1 Λ) P implies I -Λ -1 Λ ≤ dM n /n.
Then the conclusion follows from equation (2.20).

Approximation of functions

In order to prove the prior positivity of Kullback-Leibler balls around θ 0 , we need to approximate f 0 ∈ C β [-S, S] d by finite location-scale mixtures of kernels. We mostly follow the approach of de Jonge and van Zanten (2010, lemma 3.4). Nevertheless, as mentioned in de Jonge and van Zanten (2010), we shall extend f 0 defined on [-S, S] d onto a (smooth) function defined on R d to be able to approximate [START_REF] Whitney | Analytic extensions of differentiable functions defined in closed sets[END_REF] to find a function f 0 : R d → R such that f 0 ∈ C β (R d ) and D α f 0 (x) = D α f 0 (x) for all x ∈ [-S, S] d and all |α| ≤ β. Then we apply the method of de Jonge and van Zanten (2010, lemma 3.4) to f 0 . We find this approach more elegant since we do not have to assume that f 0 is defined on a larger set than the support of the covariates.

For each

α ∈ N d , let m h α := h -d x α K hI (x) dx. For α ∈ N d with |α| ≥ 1,
c α := l+k=α |l|≥1, |k|≥1 (-1) |α| α! m h l m h k m h α d k , d α := (-1) |α| α! -c α .
(2.21)

Given β > 0, h > 0 and p the largest integer strictly smaller than β, define

f β := f 0 - 1≤|α|≤p d α m h α D α f 0 .
Proposition 2.5 -Let h > 0. For any β > 0 and any function

f 0 ∈ C β [-S, S] d there is a positive constant M β such that |h -d K hI * f β (x) -f 0 (x)| ≤ M β h β for all x ∈ [ -S, S] d .
Proof. Noticing that m h α h |α| , the proof follows from the same argument as in (Shen et al., 2013, lemma 2), because f 0 (x) = f 0 (x) for all x ∈ [-S, S] d . The proposition 2.5 shows that any sufficiently regular function can be approximated by continuous location mixtures of K hI , provided h is chosen small enough and g has enough finite moments. In the sequel, we will need slightly more, that is approximating any β-Hölder continuous function by discrete mixtures of K hI ; this is done by discretizing the convolution operator in the next proposition. Compared to Ghosal and van der Vaart (2001, lemma 3.1), we need to take extra cares regarding the fact that f 0 can take negative values, and also to control the "total mass" of the mixing measure.

Proposition 2.6 -Let h > 0 be small enough and ζ = 1∨2/(τ -γτ ). There exists a discrete mixture f

(x) = N i=1 α i K hI (x -µ i ) with N h -d (log h -1 ) d(ζ-1) , µ i ∈ [-2S, 2S] d for all i = 1, . . . , N ; such that |f (x) -f 0 (x)| h β for all x ∈ [-S, S] d . Moreover N i=1 |α i | h -d , and |µ i -µ j | d ≥ h β+1 for any i = j.
Proof. Let Q be the signed measure defined by A → A f β (y)dy for any measurable

set A ⊆ R d . Let M h := (C -1 0 (β + d) log h -1
) 1/τ . To any j ∈ Z d we associate the cube B j := hM h (j + [0, 1] d ) and the signed measure

Q j such that Q j (A) := Q(A ∩ B j ) for all measurable A ⊆ R d . Let Q + j , Q - j
denote respectively the positive and negative part of the Jordan decomposition of Q j . It is a classical result from [START_REF] Tchakaloff | Formules de cubatures mécaniques à coefficients non négatifs[END_REF] that we can construct discrete (positive) measures P + j,k , P - j,k each having at most (k + d)!/(k!d!) atoms and satisfying R(x)Q ± j (dx) = R(x)P ± j (dx) for any polynomial R(x) of degree

|α| ≤ k. Let Λ h := {j ∈ Z d : |j| ≤ 1+S/(hM h )} and for any x ∈ R d let N x := {j ∈ Λ h : inf{|x -y| d : y ∈ B j } ≤ hM h }.
For the signed measure P k := j∈Λ h (P + j,k -P - j,k ) the total variation of P k satisfy the bound

|P k | ≤ j∈Λ h P + j,k + j∈Λ h P - j,k ≤ j∈Z d (P + j,k + P - j,k ) = |Q|.
Notice that |Q| < +∞ since we have

f β ∈ L 1 (R d ). Moreover, letting P j,k = P + j,k -P - j,k K hI (x -y)(Q -P k )(dy) = j / ∈Λ h B j g x -y h Q j (dy) + j∈Λ h \Nx B j g x -y h (Q j -P j,k )(dy) + j∈Nx B j g x -y h (Q j -P j,k )(
B j g x -y h (Q j -P j,k )(dy) = |α|≤k D α g(0) α! B j x -y h α (Q j -P j,k )(dy) + B j R k x -y h (Q j -P j,k )(dy), (2.23)
where

|R k (x)| ≤ sup |α|=k |D α g(0)||x| k d /k!.
The first term of the rhs of equation (2.23) vanishes by construction of P j,k . For any j ∈ N x and any y ∈ B j it holds |x-y| d ≤ 2hM h ; then using Stirling's formula and assumptions on D α g the second term of the rhs of equation (2.23) is bounded by

sup |α|=k |D α g(0)| (2eM h ) k √ 2πkk k B j |Q j -P j,k |(dy) ≤ K 1 exp {-k(1 -γ) log k + k log(2eM h )} ,
whenever j ∈ N x , for a constant K 1 depending only on f 0 , β and g. Therefore, choosing k ≥ (2eM h ) 2/(1-γ) , we deduce from equations (2.22) and (2.23) that

K hI (x -y)(Q -P k )(dy) ≤ 3|Q|h β+d + K 1 exp - 1 -γ 2 k log k . (2.24) Now if (2eM h ) 2/(1-γ) ≥ 2(β +d)/(1-γ) log h -1
set k to be the smaller integer larger than (2eM h ) 2/(1-γ) ; otherwise set k to be the larger integer greater than 2(β+d)/(1-γ) log h -1 . This yields the first part of the proposition with f (x) = h -d K hI (x -y)P k (dy) because of equation (2.24), of proposition 2.5 and because each of the P j,k has a number of atoms proportional to (log h -1 ) dζ by Tchakaloff's theorem, all in [-2S, 2S] d if h is small enough. It remains to prove the separation between the atoms of Q k . But the cost to the supremum norm of moving one µ i of h β+1 is proportional to h β by proposition 2.12. Hence we can assume that the support point of Q k are chosen on a regular grid with h β+1 separation within nodes (see also Shen et al. (2013, corollary B1)).

Kullback-Leibler property

A simple computation shows that (see for instance [START_REF] Choi | On posterior consistency in nonparametric regression problems[END_REF]) for θ 0 = (f 0 , σ 0 ) and θ = (f, σ),

K i (θ 0 , θ) = log σ σ 0 - 1 2 1 - σ 2 0 σ 2 + 1 2 |f 0 (x i ) -f (x i )| 2 σ 2 , V 2;i (θ 0 , θ) = 1 2 1 - σ 2 0 σ 2 2 + σ 4 0 σ 4 |f 0 (x i ) -f (x i )| 2 .
Therefore, for all 0 < ≤ 1/2, there exists a constant C 0 > 0 (depending only on θ 0 ) such that one has the inclusions

K n (θ 0 , ) ⊇ (f, σ) : f -f 0 2 ∞ ≤ C 0 2 , σ 0 ≤ σ ≤ σ 0 (1 + C 0 2 ) , (2.25)
hence probabilities of Kullback-Leibler balls around θ 0 are lower bounded by the probability of the sets defined in the rhs of equation (2.25). Now we state and prove the main result of this section.

Lemma 2.5 -Let f 0 ∈ C β [-S, S] d , and ζ > 1 as in proposition 2.6. Then there exists a constant C > 0, not depending on n, such that

Π(K n (θ 0 , n )) exp(-n 2 n ) for 2 n = Cn -2β/(2β+d+κ/2) (log n) 2βd(ζ-1)/(2β+d+κ/2) .
Proof. By proposition 2.6 for any h > 0 sufficiently small, there is N h -d (log h -1 ) d(ζ-1) and a function f h (x) = N j=1 α j K hI (x -µ j ) such that |f h (x) -f 0 (x)| h β for all x ∈ [-S, S] d , with α j ∈ R for all j = 1, . . . N , µ i ∈ [-2S, 2S] d for all i = 1, . . . , N , and

|µ i -µ j | d ≥ h β+1 whenever i = j. Let define E s,h := A ∈ E s : h -1 ≤ λ i (A -1 ) ≤ h -1 (1 + h β+d ), i = 1, . . . , d .
We construct a partition of E s × [-2S, 2S] d in the following way : for all j = 1, . . . , N , let U j be the closed ball of radius h β+d+1 centered at µ j (observe that these balls are disjoint), and set

V j := E s,h × U j , V c := E s × [-2S, 2S] d \U N j=1 V j . Let Q denote the set of signed measures on E s × [-2S, 2S] d satisfying Q ∈ Q ⇒ |Q(V j ) -α h,j | ≤ h β N -1 for all j = 1, . . . , N , and |Q|(V c ) ≤ h β . Notice that for any Q ∈ Q we have |Q| ≤ N j=1 |Q(V j ) -α h,j | + N j=1 |α h,j | h β + h -d h -d
because of proposition 2.6. Then for any Q ∈ Q and all x ∈ [-S, S] d , using proposition 2.12,

Es×[-2S,2S] d K A (x -µ) Q(dAdµ) -f h (x) N j=1 |Q(V j ) -α h,j | + |Q|(V c ) + N j=1 V j |K A (x -µ) -K hI (x -µ j )| |Q|(dAdµ) h β . Thus for all Q ∈ Q and all x ∈ [-S, S] d , we have | K A (x -µ) Q(dAdµ) -f 0 (x)| ≤ | K A (x -µ) Q(dAdµ) -f h (x)| + |f h (x) -f 0 (x)| ≤ K 1 h β for a constant K 1 > 0 not
depending on h. By the assumptions of equations (2.5) and (2.8) we have for any

j = 1, . . . , N αF A (E s,h )F µ (U j ) ≥ αb 1 b 4 h a 1 (β+d+1)-a 4 +a 5 (β+d) exp(-C 3 h -κ/2 ) =: K 2 h q exp(-C 3 h -κ/2 ),
where q := a 1 (β + d + 1) -a 4 + a 5 (β + d) and the constant K 2 > 0 not depending on h.

For h > 0 sufficiently small, it is clear that K 2 h q exp(-C 3 h -κ/2 ) < F (V j ) ≤ 1 for all j = 1, . . . , N . We also assume without loss of generality that K 2 h q exp(-C 3 h -κ/2 ) ≤ F (V c ) ≤ 1 and we set V N +1 := V c , α h,N +1 := 0 ; otherwise we subdivide V c onto smaller subsets for which the relation is verified. Because F is a probability measure, this can be done with a finite number of subsets not depending on h. Now let W := {σ > 0 :

σ 0 ≤ σ ≤ σ 0 (1 + C 0 2 n )} and n = C -1 0 K 1 h β . Notice that P σ (W ) ≥ K 3 2a 9 n
with a constant K 3 > 0 eventually depending on θ 0 . The sets E s,h × U j are disjoint, hence by equation (2.12) and proposition 2.11 we deduce that there is a constant

K 4 > 0 such that Π(K n (θ 0 , n )) ≥ P σ (W )Π * (Q) 2a 9 n N +1 i=1 h β N -1 e -(3+η)|α j | Γ(α(F A (E h )F µ (U j )) ≥ exp -K 4 h -(d+κ/2) (log h -1 ) d(ζ-1) , where we used that N h -d (log h -1 ) d(η-1) , N j=1 |α h,j | h -d and Γ(x)
x -1 for x > 0 sufficiently small. This concludes the proof.

Proof of theorem 2.4

As in section 2.5, the proof of theorem 2.4 consists on verifying the condition established in theorem 2.3.

Sieve construction

For constants H, M > 0 to be determined later, we define

Θ n :=        (f, σ) : f (x) = K ξ,φ (x -µ)Q(dξdµdφ), supp Q = R d × [-2S, 2S] d × [0, π/2], Q = ∞ i=1 u i δ ξi,µi,φi , n -2/a8 < σ 2 ≤ n -2/a8 (1 + M n ) n ∞ i=1 |u i | ≤ n, #{i : |u i | > n -1 , |ξ i | d ≤ e 2n 2 n } ≤ Hn 2 n / log n, ∞ i=1 |u i | 1{|ξ i | d > e 2n 2 n } ≤ M n , ∞ i=1 |u i | 1{|u i | ≤ n -1 } ≤ M n        .
In the sequel, we assume without loss of generality that the jumps of Q in the definition of Θ n are ordered so that there is no jump with |u i | > n -1 and |ξ i | d ≤ e 2n 2 n when i > Hn 2 n / log n. Moreover, we consider the following partition of Θ n . Let H n be the largest integer smaller than Hn 2 n / log n. Then for any j = (j 1 , . . . , j Hn ) ∈ {1, 2, . . .} Hn we define the slices

Θ n,j := (f, σ) ∈ Θ n : √ n(j i -1) ≤ |ξ| d < √ nj i , ∀i ≤ H n .
Lemma 2.6 -Assume that there is 0 < γ 1 < 1 such that 2 n ≥ n -γ 1 for all n large enough. Then for

H = 6(1 -γ 1 ) -1 it holds Π(Θ c n ) exp(-3n 2 n ) as n → ∞.
Proof. According to the proof of lemma 2.3, the result holds if F ξ (ξ :

|ξ d | ≥ e 2n 2 n ) 2 n exp(-3n 2 n )
for n sufficiently large. Then the conclusion follows from equation (2.13) because η > 0.

Lemma 2.7 -Let n → 0 with n 2 n → ∞ and K = 3(32 ∨ 4σ 2 0 ). Then there exists M > 0 such that it holds j N (M n , Θ n,j , ρ n ) Π(Θ n,j ) e -(KM 2 -2)n 2 n → 0.

Proof. With the same argument as in lemma 2.4, it follows from equation (2.13) that for any j ∈ {1, 2, . . .} Hn

Π(Θ n,j ) ≤ i≤Hn F ξ (ξ : |ξ| d ≥ √ n(j i -1)) ≤ b Hn 11 i≤Hn (1 + √ n(j i -1)) -2(η+1) .
Moreover, using proposition 2.7 we can find a constant η+1) .

C > 0 independent of M such that N (M n , Θ n,j , ρ n ) ≤ exp(2CHn 2 n ) i≤Hn j d-1 i when n is large. Therefore, for those n N (M n , Θ n,j , ρ n ) Π(Θ n,j ) ≤ exp Hn 2 n C + log b 11 2 log n i≤Hn j (d-1)/2 i [1 + √ n(j i -1)] -(
For n large enough we have log b 11 ≤ 2C log n ; then provided η > (d -1)/2, we can sum over j ∈ {1, 2, . . .} Hn the last expression to get

j N (M n , Θ n,j , ρ n ) Π(Θ n,j ) ≤ exp 2CHn 2 n k≥1 k (d-1)/2 [1 + √ n(k -1)] -(η+1) Hn ≤ exp 2CHn 2 n 1 + n -(η+1)/2 k≥1 k (d-1)/2-(η+1) Hn exp{3CHn 2 n },
where the last inequality holds for n sufficiently large. Now choose M > 0 satisfying KM 2 > 2 + 3CH to obtain the conclusion of the lemma.

Proposition 2.7 -For n large enough there is a constant C > 0 independent of M such that for any sequence n → 0 with n 2 n → ∞, the following holds for any j ∈ {1, 2, . . .} Hn .

log

N (M n , Θ n,j , ρ n ) ≤ CHn 2 n + (d -1) i≤Hn log j i . Proof. The proof is similar to proposition 2.4. Let R n be a (M n /n)-net of [-2S, 2S] d , ∆ n be a M n -net of {(u 1 , . . . , u Hn ) ∈ R Hn : Hn i=1 |u i | ≤ n} in the 1 -distance, S n := {σ > 0 : σ 2 = n -2/a 8 (1 + M n ) k , k ∈ N, k ≤ n}, U n be a (M n /n)-net of [0, π/2], and for all k = 1, . . . , H n , let V n,k a (M n /n)-net of {ξ ∈ R d : √ n(k -1) ≤ |ξ| d < √ nk}. Pick (f, σ) ∈ Θ n,j with f (x) = ∞ i=1 u i K ξ i ,φ i (x -µ i ). Clearly we can find u ∈ ∆ such that i≤Hn |u i -u i | ≤ M n , µ ∈ R Hn n such that |µ i -µ i | d ≤ M n /n for all i = 1, . . . , H n , φ ∈ U Hn n such that |φ i -φ i | ≤ M n /n for all i = 1, . . . , H n , ξ i ∈ V n,j i such that |ξ i -ξ i | d ≤ M n /n for all i = 1, . . . , H n , and σ ∈ S n such that | log σ -log σ| ≤ M n . Let f (x) = i≤Hn u i K ξ i , φ i (x -µ i )
denote the function built from the parameters chosen as above ; it follows

f -f 2,n ≤ i>Hn |u i | + i≤Hn |u i -u i | + i≤Hn |u i | K ξ i ,φ i (• -µ i ) -K ξ i , φ i (• -µ i ) 2,n ≤ 2M n + C i≤Hn |u i ||ξ i -ξ i | d + C i≤Hn |u i ||µ i -µ i | d + C i≤Hn |u i ||φ i -φ i | ≤ 2M (1 + 3C) n ,
for a constant C > 0 depending only on g, because of proposition 2.12. Thus a 2(1 + 3C )M n -net of Θ n,j in the distance ρ n can be constructed with ( f , σ)

as above. Recall that # R n ≤ (4Sn/(M n ) d , # ∆ n ≤ (n/(M n )) Hn , # S n = n, # U n ≤ πn/(2M n ) and # V k ≤ (n 3/2 k/(M n ) + 1) d -(n 3/2 (k -1)/(M n ) -1) d (n 3/2 /(M n )) d k d-1 , where we used u d -v d ≤ d(u -v)u -1 for v > u.
Then the end of the proof is identical to proposition 2.4.

Approximation of functions

Let ξ > 0 and m, r ≥ 1 be two positive integers. Let define the approximating kernel L m,ξ : R d → R by the expression L ξ m,r (x) := λ ξ m,r g(x) d i=1 sin 2r (mξx i )/ sin 2r (ξx i ), where λ ξ m,r > 0 is chosen so that R d L ξ m,r (x) dx = 1. Also let f 0 denote a suitable Whitney extension of f 0 from [-S, S] d to R d (see the proof of proposition 2.5). We may assume that f 0 and all its derivatives (up to order β) are zero outside [-2S, 2S] d . If it is not the case, it suffices to multiply f 0 by a smooth function that equal 1 on [-S, S] d and 0 outside [-2S, 2S] d (for instance, think about the convolution of a bump function with a proper indicator set function).

In order to achieve good order of approximation of f 0 when β is large, we construct a transformation of f 0 as follows. In the sequel we let p be the largest integer strictly smaller than β. For all multi-index α ∈ N d , we define m m,r,ξ α := R d x α L ξ m,r (x) dx. By definition of L ξ m,r , the m m,r,ξ α 's are always finite. Then we define

f β ≡ f m,r,ξ β := f 0 - 1≤|α|≤p d α m m,r,ξ α D α f 0 ,
where the coefficients (d α ) are defined in the same fashion as equation (2.21), with obvious modifications.

Proposition 2.8 -Let m, r ≥ 1 be integers. For any β > 0 and any function

f 0 ∈ C β [-S, S] d there is a constant M β > 0 such that |L r m,ξ * f β (x) -f 0 (x)| ≤ M β (log m/m) β for all x ∈ [-S, S] d if 2r ≥ p + 1 and ξ = K 0 (log m) -1
for a constant K 0 depending only on g, β and r.

Proof. First assume 0 < β ≤ 1. By assumptions on f 0 , there is M > 0 such that for all x, y ∈ R d we have

| f 0 (x) -f 0 (y)| ≤ M |x -y| β d . Then, f 0 (x) -L ξ m,r * f 0 (x) ≤ R d f 0 (x) -f 0 (y) |L ξ m,r (x -y)| dy ≤ M R d |x -y| β d |L ξ m,r (x -y)| dy.
Remark that for any τ > 0 and all u ∈ R d we have

d i=1 |u i | τ ≤ d max i=1,...,d |u i | τ ≤ d( d i=1 |u i | 2 ) τ /2 and |x -y| β d = ( d i=1 |x i -y i | 2 ) β/2 ≤ d β/2 max i=1,...,d |x i -y i | β ≤ d β/2 d i=1 |x i -y i | β . Then, because |g(x)| exp(-C 0 |x| τ d ), f 0 (x) -L ξ m,r * f 0 (x) λ ξ m,r d i=1 R d |u i | β exp -C 0 d -1 d j=1 |u j | τ d j=1 sin 2r (mξu j ) sin 2r (ξu j ) du λ ξ m,r R |u| β e -C 0 |u| τ /d sin 2r (mξu) sin 2r (ξu) du R e -C 0 |u| τ /d sin 2r (mξu) sin 2r (ξu) du d-1
.

(2.26)

We now bound the first integral of the rhs of equation (2.26). Let split the domain into three parts :

D 1 := (-1/(ξm), 1/(ξm)), D 2 := [-1/(ξm), -π/ξ] ∪ [1/(ξm), π/ξ] and D 3 := R\(D 1 ∪ D 2 ).
On D 1 and D 3 we always have sin 2 (mξu)/ sin 2 (ξu) m 2 , whereas on D 2 it holds sin 2 (mξu)/ sin 2 (ξu) 1/(ξx) 2 . Therefore,

R |u| β e -|u| τ /d sin 2 (mξu) sin 2 (ξu) du m 2r D 1 |u| β du + ξ -2r D 2 |u| β-2r du + m 2r D 3 |u| β e -C 0 |u| τ /d du =: I 1 + I 2 + I 3 .
The bounds I 1 m -β+(2r-1) ξ -(β+1) and I 2 ξ -(β+1) (1 + m -β+(2r-1) ) are obvious. Now we bound I 3 . By Markov's inequality, for any t < C 0 /d, we have

∞ π/ξ u β exp(-C 0 u τ /d) ≤ e -π/ξ ∞ 0 u β exp(-C 0 u τ /d + ut) du.
Now it is clear that I 3 m 2r exp(-π/ξ) since by assumption τ > 1 and we can choose t < C 0 /d. It follows I 3 ξ -(β+1) if ξ = K 0 (log m) -1 for a suitable constant K 0 > 0 depending only on g, β and r. The same reasoning applies to the second integral of the rhs of equation (2.26), yielding the bound

f 0 (x) -L m,ξ * f 0 (x)
λ ξ m,r m -β+d(2r-1) (log m) β+d , (2.27) whenever ξ = K 0 (log m) -1 . Hence, it remains to bound λ ξ m,r . By assumption, we have g(x) ≥ 0 for all x ∈ R d and a constant C > 0 such that g

(x) > C on a set E ⊆ [-π, π] d ; thus 1 λ ξ m,r ≥ E g(x) d i=1 sin 2r (mξx i ) sin 2r (ξx i ) dx Cm 2dr E d i=1 sin 2r (mξx i ) (mξx i ) 2r dx m d(2r-1) ξ d E d i=1 sin 2 (u i ) u 2 i du m d(2r-1) ξ d ,
where E := {mξx : x ∈ E} has non-null Lebesgue measure by assumption. Combining the last result with equation (2.27), we get the estimate

| f 0 (x) -L m,ξ * f 0 (x)| m -β (log m) β for all x ∈ R d provided ξ ≤ K 0 (log m) -1 . Now assume that β > 1.
Acting as in the previous paragraph, we can have m m,r,ξ α ≤ m -|α| (log m) |α| for all |α| ≤ p, provided 2r > p + 1 and ξ = K 0 (log m) -1 for a suitable constant K 0 > 0. Then the proof is identical to Shen et al. (2013, lemma 2).

Proposition 2.9 -Let m ≥ 1, r ≥ (β + 1)/2 be integers and ξ = K 0 (log m) -1 , with K 0 as in proposition 2.8. There exists a discrete mixture f

(x) = N i=1 α i K ξ i ,φ i (x -µ i ) with N (m log m) d and for all i = 1, . . . , N : µ i ∈ [-2S, 2S] d , ξ i ∈ [0, 2rK 0 m/ log m] d , φ i ∈ [0, π/2] ; such that |f (x)-f 0 (x)| (log m/m) β for all x ∈ [-S, S] d . Moreover N i=1 |α i | 1, and for any i = j it holds |ξ i -ξ j | d ≥ 2(log m/m) β , |µ i -µ j | d ≥ 2(log m/m) β and |φ i -φ j | ≥ 2(log m/m) β .
Proof. We rewrite L ξ m,r in a more convenient form for the sequel. Let a 0 := 1 and

a k = 2(1 -k/m) for all k = 1, . . . , m -1. Then first step is to notice that L ξ m,r (x) = m dr λ ξ m,r g(x) d i=1 m-1 k=0 a k cos(2ξkx i ) r .
From here, letting I r := {0, . . . , m -1} r and S = {-1, 1},

L ξ m,r (x) = m dr λ ξ m,r g(x) d i=1   k∈Ir a k 2 -r e∈S r cos 2ξx i r j=1 e j k j   ,
where a k := a k 1 . . . a kr , and because r j=1 cos(2ξk j x i ) = 2 -r e∈S r cos(2ξ r j=1 e j k j x i ). Notice that |a k |2 -r ≤ 1 for all k ∈ I r , and that 2| r j=1 e j k j | can take at most 1+r(m-1) values ; we denote these unique values ω j with j ∈ J := {0, . . . , r(m -1)}. Then we can rewrite,

L ξ m,r (x) = m dr λ ξ m,r g(x) d i=1   k∈J a k cos(ξω j x i )   ,
where the coefficients a k satisfy |a k | ≤ 2#(I r × S r ) ≤ 2(2m) r . Finally, for all k ∈ J d letting b k := 2 -d a k 1 . . . a k d and ω k,i := ω k i , with the same arguments as previously,

L ξ m,r (x) = m dr λ ξ m,r g(x) k∈J d e∈S d b k cos ξ d i=1 ω k,i e i x i , where |b k | ≤ (2m) dr for all k ∈ J d . Therefore, (m dr λ ξ m,r ) -1 L ξ m,r * f β (x) = k∈J d e∈S d b k R d f β (y)g(x -y) cos ξ d i=1 ω k,i e i (x i -y i ) dy = k∈J d e∈S d b k cos ξ d i=1 ω k,i e i x i R d f β (y)g(x -y) cos ξ d i=1 ω k,i e i y i dy + k∈J d e∈S d b k sin ξ d i=1 ω k,i e i x i R d f β (y)g(x -y) sin ξ d i=1 ω k,i e i y i dy.
We finish the proof by discretizing the integrals in the last equation. Obviously the proof are identical for both integrals, hence we only consider the first one. To ease notations, we set h k (x) := f β (x) d i=1 cos(ξ d i=1 ω k,i e i x i ). For any integer q ≥ 1, proceed as in the proof of proposition 2.6 to find a signed measure P k,q =: l∈L p k,l δ x k,l such that

[-2S,2S] d R(x) dP k,q (x) = [-2S,2S] d R(x) h k (x)dx for all polynomials R(x) of degree ≤ q, with #L ≤ (q + d)!/(q!d!) and l∈L |p k,l | = [-2S,2S] d |h k (x)| dx ≤ M for a positive constant M (recall that by construction of f β , we have f β ∞ < +∞, and supp f β ⊆ [-2S, 2S] d ). Then for any x ∈ R d , R d h k (y)g(x -y) dy - [-2S,2S] d g(x -y) dP k,q (y) ≤ |α|≤r |D α g(0)| α! [-2S,2S] d (x -y) α h k (y)dy - [-2S,2S] d (x -y) α dP k,q (y) + [-2S,2S] d |R q (y)| |h k (y)|dy + [-2S,2S] d |R q (y)| d|P k,q |(y), (2.28)
where |R q (y)| ≤ sup |α|=q |D α g(0)||y| q d /q!. The first term of the rhs of equation (2.28) is null by construction of P k,q . As in the proof of proposition 2.6, the two last terms of equation (2.28) are bounded by a constant multiple of exp -(1 -γ)q log q + q(1 + log(2

√ dS)) .
Then the error of approximating the integrals is o(m -β ) if q = K 1 log m for a suitable constant K 1 > 0 depending only on β and γ. Since for ξ = K 0 (log m) -1 we have,

m dr λ ξ m,r k∈J d e∈S d |b k | m dr × m -d(2r-1) × (log m) -d × #J d × m dr (log m) -d ,
the error of approximating L ξ m,r * f β by the discretized version does not exceed o(m -β ) when q = K 1 log m. The conclusion of the proposition follows from elementary manipulation of trigonometric functions and because #L q d (log m) d .

It remains to prove the separation between the atoms of the mixing measure, but this follows from proposition 2.13 with the same argument as in proposition 2.6.

Kullback-Leibler property

Lemma 2.8 -Let f 0 ∈ C β [-S, S] d . Then there exists a constant C > 0, not depending on n, such that Π(K n (θ 0 , n )) exp(-n 2 n ) for 2 n = Cn -2β/(2β+d) (log n) 2β(2d+1)/(2β+d) . Proof. Let f m (x) = N i=1 α i K ξ i ,φ i (x -µ i )
be as in proposition 2.9. For any i = 1, . . . , N define the sets

U i := {ξ ∈ R d : |ξ -ξ i | d ≤ (log m/m) β }, V i := {µ ∈ [-2S, 2S] d : |µ -µ i | d ≤ (log m/m) β } and W i := {φ ∈ [0, π/2] : |φ -φ i | ≤ (log m/m) β }.
Notice that these sets are disjoint, and for any i = 1, . . . , N we have

αF (U i × V i × W i ) |ξ i | -a 12 d (log m/m) β(a 1 +a 10 +a 13 ) (log m/m) q ,
where q := da 12 + β(a 1 + a 10 + a 13 ). Then proceed as in lemma 2.5, to find constants x+y+δ) ≥ δ e -(3+η) (x+y+δ) . Then

K 1 , K 4 > 0 such that with n = C -1 0 K 1 (log m/m) β , Π(K n (θ 0 , n )) ≥ exp -K 4 m d (log m) d+1 . δ) α-1 e -η(
P(|X -x| ≤ δ) ≥ δ e -(3+η)(x+δ) Γ(α) 2 ∞ 0 y α-1 e -(3+2η)y dy = δ e -(3+η)(x+δ) (3 + 2η) α Γ(α) ≥ δ e -(3+η)|x| e(3 + 2η) α Γ(α)
.

The proof when x < 0 is obvious.

2.B. Auxiliary results

Proposition 2.12 -Let K A (x) = g(A -1 x), and assume that for all multi-index k ∈ N d with |k| = 0, 1, 2 the mapping x → x k g(x) belongs to L 1 (R d ). Let • be the spectral norm on E. Then there is a constant C > 0 such that for all x, µ 1 , µ 2 ∈ R d and all A 1 , A 2 ∈ E arbitrary with

I -A -1 1 A 2 ∧ I -A -1 2 A 1 small enough, |K A 1 (x -µ 1 ) -K A 2 (x -µ 2 )| ≤ C I -A -1 1 A 2 ∧ C I -A -1 2 A 1 + C A -1 1 ∧ A -1 2 |µ 1 -µ 2 | d .
Proof. Starting from the triangle inequality, we have

|K A 1 (x -µ 1 ) -K A 2 (x -µ 2 )| ≤ |K A 1 (x -µ 2 ) -K A 2 (x -µ 2 )| + |K A 1 (x -µ 1 ) -K A 1 (x -µ 2 )| (2.29)
We recall that K A (x) := g(A -1 x). To bound the first term, it is enough to bound g(x) -g(A -1 1 A 2 x) for all x ∈ R d . Let (B n ) n∈N and (C n ) n∈N be two arbitrary sequences in E such that I -B -1 n C n ≤ 1/n, and let g denote the Fourier transform of g. Then, sup

x∈R d g(x) -g(B -1 n C n x) ≤ R d g(ξ) -| det(B -1 n C n )| g(B -1 n C n ξ) dξ. Remark that | det B -1 n C n | ≤ 1 + | det(I -B -1 n C n )|, and I -B -1 n C n ≤ 1/n implies that | det(I -B -1 n C n )| ≤ √ d/n d . Also, |B -1 n C n ξ| d ≤ I -B -1 n C n |ξ| d + |ξ| d ≤ (1 + 1/n)|ξ| d . It turns out that, lim n→∞ | det B -1 n C n | g(B -1 n C n ξ) = g(ξ).
We now prove that

{| det B -1 n C n | g(B -1 n C n ξ) : n ≥ 2} is dominated. By assumption, g ∈ L 1 (R d ), as well as x → x k g(x) with |k| = 1, 2. This implies that | g(ξ)| ≤ C(1+|ξ| d ) -2 for some C > 0. We already saw that | det B -1 n C n | ≤ 1 + 1/n d , and |ξ| d ≤ |B -1 n C n ξ| d + |(I -B -1 n C n )ξ| d implies |B -1 n C n ξ| d ≥ (1 -1/n)|ξ| d . Therefore, for any n ≥ 2, | det B -1 n C n | g(B -1 n C n ξ) ≤ C| det B -1 n C n | (1 + |B -1 n C n ξ| d ) 2 ≤ C(1 + 2 -d ) (1 + |ξ| d /2) 2 .
Then the dominated convergence applies, and

lim n→∞ sup x∈R d |g(x) -g(B -1 n C n x)| = 0.
The second term of the rhs of equation (2.29) is bounded above by

|A -1 1 (µ 1 -µ 2 )| d ≤ A -1 1 |µ 1 -µ 2 | d ,
using Lipshitz continuity of g. Using a symmetry argument, the conclusion of the proposition follows.

Proposition 2.13 -Let K ξ,φ (x) = g(x) cos( d i=1 ξ i x i + φ), and assume that for all multiindex k ∈ N d with |k| ≤ 1 we have

sup x∈R d |x k g(x)| < +∞ and sup x∈R d |D k g(x)|. Then there is a constant C > 0 such that for all x, µ 1 , µ 2 , ξ 1 , ξ 2 ∈ R d and all φ 1 , φ 2 ∈ [0, π/2] |K ξ 1 (x -µ 1 ) -K ξ 2 (x -µ 2 )| ≤ C|ξ 1 -ξ 2 | d + C|µ 1 -µ 2 | d + C|φ 1 -φ 2 |.
Proof. We write,

|K ξ 1 ,φ 1 (x -µ 1 ) -K ξ 2 ,φ 2 (x -µ 2 )| ≤ |K ξ 1 ,φ 1 (x -µ 1 ) -K ξ 1 ,φ 1 (x -µ 2 )| + |K ξ 1 ,φ 1 (x -µ 2 ) -K ξ 1 ,φ 2 (x -µ 2 )| + |K ξ 1 ,φ 2 (x -µ 2 ) -K ξ 2 ,φ 2 (x -µ 2 )|.
Because g has bounded first derivatives, it is Lipschitz continuous for some Lipschitz contant K > 0, then the first term of the rhs is bounded above by K|µ 1 -µ 2 | d . With the same argument, the second term is bounded by a constant multiple of g ∞ |φ 1 -φ 2 |. The last term of the rhs is easily bounded, because for all x ∈ R d :

|K ξ 1 ,φ 2 (x) -K ξ 2 ,φ 2 (x)| ≤ | cos( d i=1 ξ 1,i x i + φ 2 ) -cos( d i=1 ξ 2,i x i + φ 2 )||g(x)| ≤ d i=1 |ξ 1,i x i -ξ 2,i x i ||g(x)| ≤ d i=1 |ξ 1i -ξ 2i | 2 1/2 d i=1 |x i g(x)| 2 1/2
, where the last line holds by Hölder's inequality. Then the conclusion follows x → x k g(x)

is bounded for all |k| = 1.

Proposition 2.14 -Let g

(x) = exp(-|x| 2 d /2). Then sup x∈R d |D α g(x)| exp( 1 2 |α| log |α|) for all α ∈ N d . Proof. For any α ∈ N d , let k = |α| = d i=1 α i . When k < 2, the result is obvious. Now assume that k ≥ 2. By Fourier duality, we have for all x ∈ R d |D α g(x)| ≤ |u α g(u)| du 2 k/2 d i=1 Γ α i + 1 2 2 k/2 d i=1 2 α i + 1 α i + 1 2e α i +1 2
, where the last inequality follows from Stirling formula. Then it is clear that,

|D α g(0)| exp - k 2 - 1 2 d i=1 log(1 + α i ) + 1 2 d i=1 α i log(1 + α i ) .
The result follows because for all k ≥ 2 we have

d i=1 α i log(1+α i ) ≤ d i=1 α i log(1+k) ≤ (1/2 + log k) d i=1 α i ≤ k/2 + k log k.

Tails assumptions and posterior concentration rates for mixtures of Gaussians

Résumé

De nos jours en estimation de densité, les vitesses de contraction de la distribution a posteriori pour les mélanges en translation et en moyenne et variance de Gaussiennes ne sont connus que sous des hypothèses de queues légères; avec de meilleures vitesses obtenues pour les mélanges en translation. Il a été conjecturé par Canale et DeBlasi, mais non prouvé, que la situation devrait être inversée sous des hypothèses de queues lourdes. Cette conjecture est basée sur l'intuition qu'il n'est pas nécessaire d'obtenir un bon ordre d'approximation dans les régions avec peu d'observations (disons les queues), favorisant l'usage des mélanges en moyenne et variance qui peuvent adapter l'ordre d'approximation spatialement. Ici, nous testons l'argument précédent sur le problème de régression gaussienne avec covariables aléatoires. Bien que nous ne puissions pas encore invalider la conjecture, nous trouvons que même avec une hypothèse de queues lourdes, les mélanges en moyenne et variance semblent être moins performants que les mélanges en translation. Cependant, les preuves suggèrent l'introduction d'un mélange hybride, qui semble être toujours plus performant que les deux autres, qu'importe la nature des queues. Finalement, nous montrons que toutes les hypothèses de queues peuvent être supprimées, au prix de rendre la distribution a priori dépendente des covariables.

Abstract

Nowadays in density estimation, posterior rates of convergence for location and location-scale mixtures of Gaussians are only known under light-tail assumptions; with better rates achieved by location mixtures. It has been conjectured by Canale and DeBlasi, but not proved, that the situation should be reversed under heavy tails assumptions. The conjecture is based on the feeling that there is no need to achieve a good order of approximation in regions with few data (say, in the tails), favoring location-scale mixtures which allow for spatially varying order of approximation. Here we test the previous argument on the Gaussian errors mean regression model with random design, for which the light tail assumption is not required for proofs. Although we cannot invalidate the conjecture due to the lack of lower bound, we find that even with heavy tails assumptions, location-scale mixtures apparently perform in general worst than location mixtures. However, the proofs suggest to introduce hybrid location-scale mixtures that are find to outperform both location and location-scale mixtures, whatever the nature of the tails. Finally, we show that all tails assumptions can be released at the price of making the prior distribution covariate dependent.

Introduction

Nonparametric mixture models are highly popular in the Bayesian nonparametric literature, due to both their reknown flexibility and relative easiness of implementation, see Hjort et al. (2010) for a review. They have been used in particular for density estimation, clustering and classification and recently nonparametric mixture models have also been proposed in nonlinear regression models, see for instance de Jonge and van Zanten (2010); Wolpert et al. (2011); Naulet and Barat (2015).

There is now a large literature on posterior concentration rates for nonparametric mixture models, initiated by Ghosal and Van Der Vaart (2001); Ghosal et al. (2007a) and improved by Kruijer et al. (2010); Shen et al. (2013); Scricciolo (2014) in the context of location mixtures of Gaussian distributions and studied by Canale and De Blasi (2017) in the context of location-scale Gaussian distributions and de Jonge and van Zanten (2010) in the case of location mixture models for nonlinear regression.

Location mixture of Gaussian densities can be writen as

f σ,G (x) = R ϕ σ (x -µ)dG(µ), (3.1)
while location-scale mixtures have the form

f G (x) = R×R + ϕ σ (x -µ)dG(µ, σ). (3.2)
These models are used in the Bayesian nonparametric literature to model smooth curves, typically probability densities, by putting a prior on the mixing distribution G (and on σ for location mixtures (3.1)). The most popular prior distributions on G are either finite with unknown number of components, as in Kruijer et al. (2010) and the reknown Dirichlet Process (Ferguson (1973)) or some of its extensions. In both cases G is discrete almost surely.

In Kruijer et al. (2010) and later on in Shen et al. (2013); Scricciolo (2014) it was proved that location mixture of Gaussian distributions lead to adaptive (nearly) optimal posterior concentration rates (for L 1 metrics) over collections of Hölder types functional classes, in the context of density estimation for independently and identically distributed random variables. Contrarywise, in Canale and De Blasi (2017), suboptimal posterior concentration rates are derived and the authors obtain rates that are at best n -β/(2β+2) up to a log n term in place of n -β/(2β+1) . These results are obtained under strong assumptions on the tail of the true density f 0 , since it is assumed that f 0 (x) e -c|x| τ when x goes to infinity, for some positive c, τ .

In Canale and De Blasi (2017), the authors suggest that location-scale mixtures might lead to suboptimal posterior concentration rates, for light tail distributions but might be more robust to tails, since the rate n -β/(2β+2) is the minimax estimation rate for density estimation with regularity β, under the L 2 loss, see Reynaud-Bouret et al. (2011); Goldenshluger and Lepski (2014).

The question thus remains open as to how robust to tails mixtures of Gaussian distributions (either location or location-scale) are.

Interestingly in Bochkina and Rousseau (2016), much weaker tail constraints are necessary to achieve the minimax rate n -β/(2β+1) , for estimating densities on R + using mixtures of Gamma distributions. The authors merely require that F 0 allows for a moment of order striclty greater than 2. However in Bochkina and Rousseau (2016) as well as in Kruijer et al. (2010); Shen et al. (2013); Scricciolo (2014), the smoothness functional classes are non standard and roughly correspond to requiring that the logdensity is locally Hölder, which blurs the understanding of the robustness of Gaussian mixtures to tails. These smoothness conditions are required to ensure that the density f 0 can be approximated by a mixture f σ,G where G is a probability measure in terms of Kullback-divergence. Hence to better understand the ability of mixture models to capture heavy tails we study their use in nonparametric regression models:

Y i = f (X i ) + i , i i.i.d ∼ N (0, s 2 ), i = 1, . . . , n, X 1 , . . . , X n i.i.d ∼ Q 0 , f ∈ L 2 (Q 0 ). (3.3)
The parameter is f with prior distribution denoted by Π. We assume that s is known, which is just a matter of convenience for proofs. All the results of the paper can be translated to the case s unknown using the same methodology as [START_REF] Salomond | Bayesian testing for embedded hypotheses with application to shape constrains[END_REF] or Naulet and Barat (2015). Our aim is to study posterior concentration rates around the true regression function f 0 defined by sequences n converging to zero with n and such that

Π n -1 n i=1 |f (x i ) -f 0 (x i )| 2 ≤ 2 n | y n , x n = 1 + o p (1), (3.4)
under the model f 0 . By analogy to the case of density estimation of Reynaud-Bouret et al. ( 2011) and Goldenshluger and Lepski (2014) we assume that f 0 ∈ L 1 and belongs to a Hölder ball with smoothness β. The tail condition are then on the design distribution and written as R |x| p dQ 0 (x) < +∞, p ≥ 0, and our aim is to study the posterior concentration rate (3.4) for both location and location-scale mixtures. We show in section 3.2, that in most cases location mixtures have a better posterior concentration rate than location-scale mixtures and unless p goes to infinity the posterior concentration rates is not as good as the usual n -β/(2β+1) . This rate is suboptimal for light tail design points, since in this case the minimax posterior concentration rate is given by n -β/(2β+1) . To improve on this rate we propose a new version of location-scale mixture models, which we call the hybrid location-scale mixture and we show that this nonparametric mixture model leads to better posterior concentration rates than the location mixture (and thus than the location-scale mixture). All these results are up to log n terms. The results are summarized in table 3.1 which displays the value q defined by 2 n = n -q . Although the results are presented in the regression model, we believe that similar phenomena should take place in the density estimation problem.

The main results with the description of the three types of prior models and the associated posterior concentration rates are presented in section 3.2. Proofs are presented in section 3.3 and some technical lemmas are proved in the appendix.

Notations

We call P f (• | X) the distribution of the random variable Y | X under the model (3.3), associated with the regression function f . Given (X 1 , . . . , X n ), P n f (• | X 1 , . . . , X n ) stands for the distribution of the random vector (Y 1 , . . . , Y n ) of independent random variables Table 3.1: Summary of posterior rates of convergence for different types of mixtures. The rates are understood to be in the form 2 n = n -q , up to powers of log n factors, where q is given below.

0 < p < 2 p ≥ 2 0 < p ≤ 2β β + 1 2β β + 1 < p ≤ 2β p > 2β 0 < p ≤ 2β β + 1 2β β + 1 < p ≤ 2β p > 2β Location 2β 3β + 1 2β 3β + 1 2β 3β + 1 2β 2β + 1 + 2β/p 2β 2β + 1 + 2β/p 2β 2β + 1 + 2β/p Location-scale 2β 3β + 2 2β 2β + 1 + 2β/p 2β 2β + 2 2β 3β + 2 2β 2β + 1 + 2β/p 2β 2β + 2 Hybrid 2β 3β + 1 p p + 1 2β 2β + 1 2β 3β + 1 p p + 1 2β 2β + 1 Y j ∼ P f (• | X j )
. Also, for any random variable X with distribution P , and any function g, P g(X) denote the expectation of g(X).

For any α > 0, we let SGa(α) denote the symmetric Gamma distribution with parameter α; that is X ∼ SGa(α) has the distribution of the difference of two independent Gamma random variables with parameters (α, 1).

For any finite positive measure α on the measurable space (X, X ), let Π α denote the symmetric Gamma process distribution with parameter α (Wolpert et al., 2011;Naulet and Barat, 2015); that is, an M ∼ Π α is a random signed measure on (X, X ) such that far any disjoints B 1 , . . . , B k ∈ X the random variables M (B 1 ), . . . , M (B k ) are independent with distributions SGa(α(B i )), i = 1, . . . , k.

For any β > 0, we let C β denote the Hölder space of order β; that is the set of all functions f : R → R that have bounded derivatives up to order m, the largest integer smaller than β, and such that the norm f

C β := sup k≤m sup x∈R |f (k) (x)| + sup x =y |f (m) (x) -f (m) (y)|/|x -y| β-m is finite.
For 1 ≤ p < ∞ we let L p be the space of function for which the norm f p p := |f (x)| p dx is finite; and by L ∞ we mean the space of functions for which f ∞ := sup x∈R |f (x)| is finite. For 0 ≤ p, q ≤ ∞ and functions f ∈ L p , g ∈ L q , we write f * g the convolution of f and g, that is f * g(x) := f (x -y)g(y) dy for all x ∈ R. Moreover, we'll use repeatedly Young's inequality which state that f * g r ≤ f p g q , with 1/p + 1/q = 1/r + 1.

If f ∈ L 1 , then we define f as the (L 1 ) Fourier transform of f ; that is f (ξ) := f (x)e -iξx dx for all ξ ∈ R. Moreover, if f ∈ L 1 , then the inverse Fourier transform is well-defined and f (x) = (2π) -1 f (ξ)e ixξ dξ. Also, we denote by S the Schwartz space; that is the space of infinitely differentiable functions f : R → R for which |x r f (k) (x)| < +∞ for all r > 0 and all k ∈ N. Then S ⊂ L 1 , and it is well known that the Fourier transform maps S onto itself, thus the Fourier transform is always invertible on S. We note f r,k = sup{|x| r |f (k) (x)|, x ∈ R} for any f ∈ S.

For two real numbers a, b, the notation a∧b stand for the minimum of a and b whereas a ∨ b stand for the maximum. Similarly, given two real valued functions f, g the function f ∧ g is the function which at x assigns the minimum of f (x) and g(x) and f ∨ g has obvious definition. Throughout the paper C denotes a generic constant.

Inequalities up to a generic constant are denoted by and .

Posterior convergence rates for Symmetric Gamma mixtures

In this section we present the main results of the paper. We first present the three types of priors that are studied; i.e. location mixtures, location-scale mixtures and hybrid location-scale mixtures and for each of these families of priors we provide the associated posterior concentration rates.

Recall that we consider observations (Y i , X i ) n i=1 independent and identically distributed according to model (3.3) and we note

y n = (Y 1 , • • • , Y n ) and x n = (X 1 , • • • , X n ).
We denote the prior and the posterior distribution on f by Π(•) and Π(• | y n , x n ) respectively.

Family of priors Location mixtures of Gaussians

A symmetric Gamma process location mixture of Gaussians prior Π is the distribution of the random function f (x) := ϕ((x -µ)/σ) dM (µ) where σ ∼ G σ and M ∼ Π α , with α a finite positive measure on R, G σ a probability measure on (0, ∞) and ϕ(x) := e -x 2 /2 for all x ∈ R.

We restrict our discussion to priors for which the following conditions are verified. We assume that there are positive constants a 1 , a 2 , a 3 and b

1 , b 2 , b 3 , b 4 such that G σ satisfies for x ≥ 1 G σ (σ > x) exp(-a 1 x b 1 ) (3.5) G σ (σ ≤ 1/x) exp(-a 2 x b 2 ) (3.6) G σ x -1 ≤ σ ≤ x -1 (1 + t) x b 3 t b 4 exp(-a 3 x), ∀t ∈ (0, 1). (3.7)
We let α := αG µ for a positive constant α > 0 and G µ a probability distribution on R.

We assume that there are positive constants b 5 , b 6 such that G µ satisfies for all

x ∈ R G µ (|µ -x| ≤ t) t b 5 (1 + |x|) -b 6 , ∀t ∈ (0, 1). (3.8)
The heavy tail condition on G µ is required to not deteriorate the rate of convergence when Q 0 is heavy tailed. Notice that equation (3.5) forbids the use of the classical inverse-Gamma distribution as prior distribution on σ because of its heavy tail. In fact, it is always possible to weaken equation (3.5) to allow for Inverse-Gamma distribution (see Canale and De Blasi (2017); Naulet and Barat (2015)) but it complicates the proofs with no contribution to the subject of the paper. We found that among the usual distributions the inverse-Gaussian is more suitable for our purpose since it fulfills all the equations (3.5) to (3.7), as shown in proposition 3.1. We recall that the inverse-Gaussian distribution on (0, ∞) with parameters a > 0, b > 0 has density with respect to Lebesgue measure

f (x; a, b) := b 2πx 3 1/2 exp - b(x -a) 2 2a 2 x , ∀x > 0,
and f (x; a, b) = 0 elsewhere. Proof. It suffices to write, for any x ≥ 1

G σ (σ > x) ≤ b 2πx 3 1/2 ∞ x exp - b(t -a) 2 2a 2 t dt ≤ b 2π 1/2 exp b a - b 2 ∞ x exp - bt 2a 2 dt.
Also, for any x ≥ 1

G σ (σ ≤ 1/x) ≤ b 2π 1/2 1/x 0 t -3/2 exp - b(t -a) 2 2a 2 t dt ≤ b 2π 1/2 e b/a 1/x 0 t -3/2 e -b/(2t) dt ≤ 216(b √ e) -3 b 2π 1/2 e b/a 1/x 0 e -b/(4t) dt.
Finally, for any x ≥ 1 and 0 < t < 1,

G σ x -1 ≤ σ ≤ x -1 (1 + t) ≥ b 2π 1/2 e b/a-b/a 2 x -1 (1+t)
x -1 e -b/(2t) dt.

Location-scale mixtures of Gaussians

A symmetric Gamma process location-scale mixture of Gaussians prior Π is the distribution of the random function f (x) := ϕ((x -µ)/σ) dM (σ, µ) where M ∼ Π α , with α a finite positive measure on (0, ∞) × R and ϕ(x) := e -x 2 /2 for all x ∈ R. We focus the attention of the reader on the fact that althought we use the same notations (i.e. Π, α) as the previous section, these are different distributions and in the sequel we pay attention as making the context clear enough to avoid confusions. We restrict our discussion to priors for which α := αG σ × G µ , with α > 0 and G σ , G µ satisfying the same assumptions as in section 3.2.1.

Hybrid location-scale mixtures of Gaussians

The proof of the results given in the two preceeding sections suggests that neither location or location-scale mixtures can achieve the optimal rates, whatever the nature of the tails of Q 0 . We show that we can get better upper bounds by introducing hybrid mixtures.

By a hybrid location-scale mixtures of Gaussians, we mean the distribution Π of the random function f (x) := ϕ((x -µ)/σ) dM (σ, µ), where M ∼ Π α , with α = αP σ × G µ , α > 0, P σ ∼ Π σ and G µ a probability measure satisfying equation (3.8). Here Π σ is a prior distribution on the space of probability measures (endowed with Borel σ-algebra). We now formulate conditions on Π σ that are the random analoguous to equations (3.5) and (3.6). For the same constants a 1 , a 2 , b 1 , b 2 as in section 3.2.1, we consider the existence of positive constants a 4 , a 5 such that Π σ satisfies for x > 0 large enough

Π σ P σ : P σ (σ > x) ≥ exp(-a 1 x b 1 /2)
exp(-a 4 x b 1 ), (3.9)

Π σ P σ : P σ (σ < 1/x) ≥ exp(-a 2 x b 2 /2) exp(-a 5 x b 2 ). (3.10)
As a replacement of equation (3.7), we assume that for all r ≥ 1 there are constants a 6 , b 7 such that for any positive integer J large enough

Π σ ∩ J j=0 {P σ : P σ [2 -j , 2 -j (1 + 2 -Jr )] ≥ 2 -J } exp(-a 6 J b 7 2 J ). (3.11)
Equations (3.9) to (3.11) are rather restrictive and it is not clear a priori whether or not such distribution exists. For example, if P σ is chosen to be almost-surely an Inverse-Gaussian distribution with parameters b, µ then equation (3.11) is not satisfied. However, we now show that under conditions on the base measure, Π σ can be chosen as a Dirichlet Process, hereafter referred to as DP.

We recall that if Π σ is a Dirichlet Process distribution with base measure α σ G(•) on (0, ∞) (Ferguson, 1973), then P σ ∼ Π σ is a random probability measure on (0, ∞) such that for any Borel measurable partition A 1 , . . . , A k of (0, ∞), the joint distribution of the random variables

P σ (A 1 ), . . . , P σ (A k ) is the k-variate Dirichlet distribution with parameters α σ G(A 1 ), . . . , α σ G(A k ).
Proposition 3.2 -Let α σ > 0, G σ a probability measure on (0, ∞) satisfying the same assumptions as in equations (3.5) to (3.7), and Π σ be a Dirichlet Process with base measure α σ G σ (•). Then Π σ satisfies equations (3.9) to (3.11) with constants a 4 = a 1 , a 5 = a 2 , a constant a 6 > 0 eventually depending on r, and b 7 = 0.

Proof. We first prove equation (3.9). It follows from the definition of the DP that P σ (x, ∞) has Beta distribution with parameters α σ G σ (x, ∞) and α σ (1 -G σ (x, ∞)), then by Markov's inequality

Π σ P σ : P σ (x, ∞) ≥ t ≤ G σ (x, ∞) t .
Likewise, if t = exp(-a 1 x b 1 /2) and G σ satisfies equations (3.5) to (3.7), the conclusion follows. The same steps with G σ (0, 1/x) give the proof of equation (3.10). It remains to prove equation (3.11). Let r ≥ 1 and define V j,r := {σ : 2

-j ≤ σ ≤ 2 -j (1 + 2 -Jr )} for any integer 0 ≤ j ≤ J. For all r ≥ 1 the V j,r 's are disjoint. Set V c r := ∪ J j=0 V c j,r . If α σ G σ (V c r ) ≤ 1 let V J+1,r = V c r and M = 1 ; otherwise split V c r into M > 1 disjoint subsets V c 1,r , . . . V c M,r such that exp(-2 J ) ≤ α σ G σ (V c k,r ) ≤ 1 for all k = 1, . . . , M and set V J+1,r = V c 1,r , V J+2,r = V c 2,r , . . . , V J+M,r = V c M,r (since G σ (0, ∞) = 1
this can be done with a number M independent of J). For J large enough (so that (J + M )2 -J+1 < 1), acting as in Ghosal et al. (2000, lemma 6.1), it follows

Π σ P σ : P σ [2 -j , 2 -j (1 + 2 -Jr ] ≥ 2 -J ∀ 0 ≤ j ≤ J ≥ Γ(α σ )2 -J(J+M ) J+M j=0 Γ(α σ G σ (V j,r )) , Also, α σ G σ (V j,r ) ≤ 1 implies Γ(α σ G σ (V j,r )) ≤ 1/(α σ G σ (V j,r )), hence Π σ P σ : P σ [2 -j , 2 -j (1 + 2 -Jr ] ≥ 2 -J ∀ 0 ≤ j ≤ J ≥ Γ(α σ )α J+M +1 σ 2 -J(J+M ) J+M j=0 G σ (V j,r ).
Since M does not depend on J, one can find a constant C > 0 such that

Π σ P σ : P σ [2 -j , 2 -j (1 + 2 -Jr ] ≥ 2 -J ∀ 0 ≤ j ≤ J ≥ Γ(α σ ) exp    -CJ 2 + J j=0 log G σ (V j,r ) + J+M j=J+1 log G σ (V j,r )    .
By construction, the second sum in the rhs of the last equation is lower bounded by -M 2 J , whereas if G σ satisfies equations (3.5) to (3.7), the first sum is lower bounded by -C 2 J for a constant C > 0 eventually depending on r. Then the proposition is proved.

Posterior concentration rates under the mixture priors

We let Π(• | y n , x n ) denote the posterior distribution of f ∼ Π based on n observations (X 1 , Y 1 ), . . . , (X n , Y n ) modelled as in section 3.1. Let ( n ) n≥1 be a sequence of positive numbers with lim n n = 0, and

d n denote the empirical L 2 distance, that is nd n (f, g) 2 = n i=1 |f (X i ) -g(X i )| 2 .
The following theorem is proved in Section section 3.3 .

Theorem 3.1 -Consider the model (3.3), and assume that f 0 ∈ L 1 ∩C β and Q 0 |X| p < +∞.

Then there exist a constant C > 0 and t > 0 depending only on f 0 and Q 0 such that

• If the prior Π is the symmetric Gamma location mixture of Gaussians as defined in section 3.2.1

Π d n (f, f 0 ) 2 > Cn -2β/(3β+1) (log n) t | y n , x n = o p (1)
when 0 < p ≤ 2, and

Π d n (f, f 0 ) 2 > Cn -2β/(2β+1+2β/p) (log n) t | y n , x n = o p (1)
when p > 2. • If the prior Π is the symmetric Gamma location-scale mixture of Gaussians defined in section 3.2.1

Π d n (f, f 0 ) 2 > C[n -2β/(3β+2) ∧ n -2β/(2β+1+2β/p) ](log n) t | y n , x n = o p (1)
when 0 < p ≤ 2β, and

Π d n (f, f 0 ) 2 > Cn -β/(β+1) (log n) t | y n , x n = o p (1),
when p > 2β.

• If the prior Π is the hybrid symmetric Gamma location-scale mixture of Gaussians defined in section 3.2.1

Π d n (f, f 0 ) 2 > C[n -2β/(3β+1) ∧ n -p/(p+1) ](log n) t | y n , x n = o p (1), when 0 < p ≤ 2β or Π d n (f, f 0 ) 2 > Cn -2β/(2β+1) (log n) t | y n , x n = o p (1), when p > 2β .
The upper bounds on the rates in the previous paragraph are no longer valid when p = 0. Indeed the constant C > 0 depends on p and might not be definite if p = 0 ; the reason is to be found in the fact that C heavily depends on the ability of the prior to draw mixture component in regions of observed data, which remains concentrated near the origin when p > 0. In section 3.2.3, we overcome this issue by making the prior covariate dependent ; this allows to derive rates under the assumption p = 0 (no tail assumption).

Relaxing the tail assumption : covariate dependent prior for location mixtures

Although the rates derived in section 3.3 do not depend on p > 0 when p is small, the assumption Q 0 |X| p < +∞ is crucial in proving the Kullback-Leibler condition. Indeed, this condition ensures that the covariates belong to a set X n which is not too large, which allows us to bound from below the prior mass of Kullback-Leibler neighbourhoods of the true distribution. Surprisingly, it seems very difficult to get rid of this assumption under a fully Bayesian framework without fancy assumptions, while making the prior covariates dependent allows to drop all tail conditions on Q 0 . Doing so, we can adapt to the tail behaviour of Q 0 , as shown in the following theorem, which is an adaptation of the general theorems of Ghosal et al. (2007b). For convenience, in the sequel we drop out the superscript n and we write x, y for x n , y n , respectively. For > 0 and any subset A of a metric space equipped with metric d, we let N ( , A, d) denote the -covering number of A, i.e. N ( , A, d) is the smallest number of balls of radius needed to cover A.

Theorem 3.2 -Let Π x be a prior distribution that depends on the covariate vector x,

0 < c 2 < 1/4 and n → 0 with n 2 n → ∞. Suppose that F n ⊆ F is such that Q n 0 Π x (F c n ) exp(-1 2 (1 + 2c 2 )n 2 n ) and log N ( n /18, F n , d n ) ≤ n 2 n /4 for n large enough. If for any x ∈ R n it holds Π x (f : d n (f, f 0 ) ≤ s n ) exp(-c 2 n 2 n ), then for all M > 0 we have Π x (f : d n (f, f 0 ) > M n | y, x) = o p (1).
We apply theorem 3.2 to symmetric Gamma process location mixtures of Gaussians in the following way. Let Q n

x denote the empirical measure of the covariate vector x. Given a probability density function g, we let G x the probability measure which density is z → g(z -x i ) dQ n x (x). Corollary 3.1 -Then we let Π x be the distribution of the random function f (x) := ϕ((xµ)/σ) dM (µ), where σ ∼ G σ and M ∼ Π α with α = αG x for some α > 0. Assume that G σ satisfies equations (3.5) to (3.7) and that there exists a constant b 8 > 0 such that

sup x∈R n G x (µ : |µ -s| ≤ t) t b 8 for all 0 < t, s ≤ 1. Then Π x (f : d n (f, f 0 ) > M n | y, x) = o p (1) with 2 n n -2β/(3β+1) (log n) 2-2β/(3β+1) .
To prove corollary 3.1, note that neither the proof of lemma 3.4 or lemma 3.5 involve the base measure α (indeed, it only involves α); thus we can use the sieve F n constructed in section 3.4.1. To apply theorem 3.2 it is then sufficient to prove that for all x ∈ R n

Π x (f : d n (f, f 0 ) ≤ s n ) exp(-c 2 n 2 n ). (3.12)
This is done in lemma 3.1.

Lemma 3.1 -Assume that there is a constant b 8 > 0 such that sup x∈R n G x (µ : |µ -s| ≤ t) t b 8 for all 0 < t, s ≤ 1. Also assume that G σ satifies equations (3.5) to (3.7). Then equation (3.12) holds for the symmetric Gamma location mixture of Gaussians with base measure 3β+1) for an appropriate constant C > 0. The proof of lemma 3.1 is given in section 3.B.

αG x if 2 n ≤ Cn -2β/(3β+1) (log n) 2-2β/(

Proofs

To prove theorem 3.1 we follow the lines of Ghosal et al. (2000); Ghosal and Van Der Vaart (2001); Ghosal et al. (2007a). Namely we need to verify the following three conditions

• Kullback-Leibler condition : For a constant 0 < c 2 < 1/4, Π(KL(f 0 , n )) ≥ e -c 2 n 2 n , ( 3.13)
where

KL(f 0 , n ) := f : 1 2s 2 |f 0 (x) -f (x)| 2 dQ 0 (x) ≤ 2 n .
• Sieve condition : There exists

F n ⊂ F such that Π(F c n ) ≤ e -1 2 (1+2c 2 )n 2 n (3.14)
• Tests : Let N ( n /18, F n , d n ) be the logarithm of the covering number of F n with radius n /18 in the d n (•, •) metric.

N ( n /18, F n , d n ) ≤ n 2 n 4 . (3.15)
The Kullback-Leibler condition is proved by defining an approximation of f by a discrete mixture under weak tail conditions. Although the general idea is close to Kruijer et al. (2010) or Scricciolo (2014), the construction remains quite different to be able to handle various tail behaviours. This is detailed in the following section.

Approximation theory

To describe the approximation of f 0 by a finite mixture, we first define a few notations.

Let χ be a C ∞ function that equals 1 on [-1, 1] and 0 outside [-2, 2] c (think for instance as the convolution of 1 Fourier transform satisfies η(ξ) = χ(ξ)/ ϕ(ξ) for all ξ ∈ [-2, 2] and η(ξ) = 0 elsewhere. For two positive real numbers h and σ, we define the kernel K h,σ : R × R → R such that

[-1,1] with x → exp(-1/(1 -x 2 )) 1 [-1,1] (x)).
K h,σ (x, y) := h σ k∈Z ϕ x -hσk σ η y -hσk σ , ∀(x, y) ∈ R × R.
For a measurable function f we introduce the operator associated with the kernel :

K h,σ f (x) = K h,σ (x, y)f (y) dy.
The function K h,σ f will play the role of an approximation for the function f , and we will evaluate how this approximation becomes close to f given h and σ sufficiently close to zero. More precisely, we will prove that, when choosing h appropriately, f can be approximated by K h,σ (χ σ * f 0 ) to the order σ β . Moreover K h,σ (χ σ * f 0 ) can be written as

k∈Z u k ϕ((x -µ k )/σ)).
In a second step we approximate K h,σ (χ σ * f 0 ) by a truncated version of it, retaining only the k's such that |u k | is large enough and |µ k | not too large. In the case of location-scale and hybrid location-scale mixtures we consider a modification of this approximation to control better the number of components for which σ needs to be small. We believe that these constructions have interest in themselves. In particular they shed light on the relations between Gaussian mixtures and wavelet approximations.

These approximation properties are presented in the following two Lemmas which are proved in section 3.A:

Lemma 3.2 -There is C > 0 depending only on β such that for any f 0 ∈ L 1 ∩ C β and any σ > 0 we have |χ σ * f 0 (x) -f 0 (x)| ≤ C f C β σ β for all x ∈ R. Lemma 3.3 -Let f σ := χ σ * f 0 and h ≤ 1. Then there is a universal constant C > 0 such that |K h,σ f σ (x) -f σ (x)| ≤ C f 0 1 σ -1 e -4π 2 /h 2 for all x ∈ R.
We now present the approximation schemes in the context of location mixtures.

Construction of the approximation under location mixtures

Let 0 < σ ≤ 1 and h σ log σ -1 := 2π √ β + 1. Then combining the results of lemma 3.2 and lemma 3.

3 we can conclude that |K hσ,σ (χ σ * f 0 )(x) -f 0 (x)| σ β . Now we define the coefficients u k , k ∈ Z so that K hσ,σ (χ σ * f 0 )(x) =: k∈Z u k ϕ x -µ k σ , ∀k ∈ Z,
where

µ k := h σ σk for all k ∈ Z. Let define Λ := k ∈ Z : |u k | > σ β , |µ k | ≤ σ -2β/p + σ 2(β + 1) log σ -1 , U σ := {σ : σ ≤ σ ≤ σ(1 + σ β )}, and for all k ∈ Λ we define V k := {µ : |µ -µ k | ≤ σ β+1 } and V = ∪ k∈Λ V k .
We also denote

M σ := M signed measure on R : |M (V k ) -u k | ≤ σ β , ∀k ∈ Λ : |M |(V c ) ≤ σ β ,
and for any M ∈ M σ , we write f M,σ (x) := ϕ((x -µ)/σ) dM (µ).

Proposition 3.3 -For σ > 0 small enough, it holds |Λ| σ -(β+1) ∧ h -1 σ σ -(2β/p+1) .
Proof. Because there is a separation of h σ σ between two consecutive µ k , it is clear that |Λ| ≤ 2h -1 σ σ -(2β/p+1) . Moreover, from proposition 3.9 we have the following estimate.

f 0 1 σ -1 k∈Z |u k | ≥ k∈Λ |u k | ≥ σ β |Λ|.
Proposition 3.4 -For all x ∈ R, all σ > 0 small enough and all

M ∈ M σ it holds |f M,σ (x) -f 0 (x)| h -1 σ . Proof. For any M ∈ M σ , we have that |f M,σ (x) -f 0 (x)| ≤ |f M,σ (x)| + f 0 ∞ . But, with I ≡ I(x) := {k ∈ Z : |x -µ k | ≤ 2σ}, f M,σ (x) = k∈Λ∩I V k ϕ x -µ σ dM (µ) + k∈Λ∩I c V k ϕ x -µ σ dM (µ) + V c ϕ x -µ σ dM (µ). (3.16)
Clearly the last term of this last expression is bounded above by ϕ ∞ σ β . For the second term, we have for any

µ ∈ V k with k ∈ I c that |x -µ| ≥ |x -µ k | -|µ -µ k | ≥ |x -µ k |/2.
Then the second term of the rhs of equation (3.16) is bounded above by sup

k∈Λ∩I c |M |(V k ) k∈Z ϕ x -h σ σk σ .
Proceeding as in the proof of lemma 3.9, we deduce that the series in the last expression is bounded above by a constant times 1/h σ , whereas proposition 3.9 and Young's inequality

yields |M |(V k ) ≤ |M (V k ) -u k | + |u k | σ β + χ σ * f 0 ∞ ≤ σ β + χ 1 f 0 ∞ . Therefore the second term of the rhs in equation (3.16) is bounded by a constant multiple of h -1 σ . Regarding the first term in equation (3.16), it is bounded by ϕ ∞ |I| sup k∈Λ |M |(V k ), which is in turn bounded by h -1
σ times a constant.

Proposition 3.5 -For all σ > 0 small enough, all x ∈ R with |x| ≤ σ -2β/p and all

M ∈ M σ it holds |f M,σ (x) -f 0 (x)| h -2 σ σ β . Proof. We define A σ (β) := 2 log |Λ| + 2(β + 1) log σ -1 . Then for any M ∈ M σ , letting J ≡ J (x) := {k ∈ Z : |x -µ k | ≤ 2σA σ (β)}, we may write f M,σ (x) -K hσ,σ (χ σ * f 0 )(x) = k∈Λ∩J V k ϕ x -µ σ -ϕ x -µ k σ dM (µ) + k∈Λ∩J [M (V k ) -u k ] ϕ x -µ k σ + k∈Λ∩J c V k ϕ x -µ σ dM (µ) - k∈Λ∩J c u k ϕ x -µ k σ - k∈Λ c u k ϕ x -µ k σ + V c ϕ x -µ σ dM (µ) := r 1 (x) + r 2 (x) + r 3 (x) + r 4 (x) + r 5 (x) + r 6 (x). (3.17)
With the same argument as in proposition 3.3, we deduce that

|J | ≤ 2h -1 σ A σ (β). The same proposition implies A σ (β) log σ -1 . Recalling that |M |(V k ) 1+ χ 1 f 0 ∞ for all k ∈ Λ and all M ∈ M σ , it follows from proposition 3.11 that |r 1 (x)| A σ (β)h -1 σ σ β . From the definition of M σ , it comes |r 2 (x)| ≤ ϕ ∞ |J |σ β ≤ 2 ϕ ∞ A σ (β)h -1 σ σ β . When- ever k ∈ Λ ∩ J c and µ ∈ V k , it holds |x -µ| ≥ |x -µ k | -|µ -µ k | ≥ σA σ (β). Therefore, |r 3 (x)| ϕ(A σ (β))|Λ| σ β+1
. With the same argument, proposition 3.9 and Young's inequality we get

|r 4 (x)| χ σ * f 0 ∞ ϕ(2A σ (β))|Λ| ≤ χ 1 f 0 ∞ σ β . Regarding r 5 , we rewrite Λ c = Λ c 1 ∪ Λ c 2 , with Λ c 1 := {k ∈ Z : |u k | ≤ σ β } and Λ c 2 := {k ∈ Z : |µ k | > σ -2β/p + σ 2(β + 1) log σ -1 }. Then, |r 5 (x)| ≤ k∈Λ c 1 |u k | ϕ x -µ k σ + k∈Λ c 2 |u k | ϕ x -µ k σ ≤ σ β sup x∈R k∈Z ϕ x -µ k σ + k∈Λ c 2 |u k | ϕ x -µ k σ . (3.18)
The first term of the rhs of equation (3.18) is bounded by a multiple constant of h -1 σ σ β , with the same argument as in the proof of lemma 3.9. By definition of

Λ c 2 , |x-µ k | ≥ σ 2(β + 1) log σ -1 when k ∈ Λ c
2 and |x| ≤ σ -2β/p . This implies, together with proposition 3.9 and Young's inequality, that the second term of the rhs of equation (3.18) is bounded by a constant multiple of

σ β+1 k∈Z |u k | χ σ * f 0 1 σ β ≤ χ 1 f 0 1 σ β for all |x| ≤ σ -2β/p . Finally, we have the trivial bound |r 6 (x)| ≤ ϕ ∞ |M |(V c ) ≤ ϕ ∞ σ β .

Construction of the approximation under location-scale and hybrid location-scale mixtures

Let σ 0 := 1 and define recursively σ j+1 := σ j /2 for any j ≥ 0. Let ∆ 0 := f 0 -χ σ 0 * f 0 , and define recursively ∆ j+1 := ∆ j -χ σ j+1 * ∆ j , for any j ≥ 0.

The general idea of the construction is that |∆ j | σ β j , as shown in proposition 3.10 in appendix, and that similarly to wavelet decomposition, we approximate a function f 0 Hölder β by

f 1 := K 0 (χ σ 0 * f 0 ) + J j=1 K j (χ σ j * ∆ j-1 ).
where J ≥ 1 is a large enough integer, h J √ J := 2π/ √ β log 2, and

K j := K h J ,σ j . By induction, we get that ∆ j = ∆ 0 -j-1 l=0 χ σ l+1 * ∆ l . It follows, f 1 -f 0 = K 0 (χ σ 0 * f 0 ) -f 0 + J j=1 K j (χ σ j * ∆ j-1 ) = ∆ J + K 0 (χ σ 0 * f 0 ) -χ σ 0 * f 0 + J j=1 K j (χ σ j * ∆ j-1 ) -χ σ j * ∆ j-1 .
Therefore, from lemma 3.3 and proposition 3.10 and Young's inequality, the error of

approximating f 0 by f 1 is |f 1 (x) -f 0 (x)| ≤ |∆ J | + |K 0 (χ σ 0 * f 0 ) -χ σ 0 * f 0 | + J j=1 |K j (χ σ j * ∆ j-1 ) -χ σ j * ∆ j-1 | f 0 C β σ β J + χ σ 0 * f 0 1 σ -1 0 e -4π 2 /h 2 J + e -4π 2 /h 2 J J j=1 χ σ j * ∆ j-1 σ -1 j f 0 C β σ β J + f 1 e -4π 2 /h 2 J + f 0 1 e -4π 2 /h 2 J J j=1 2 j f 0 C β σ β J + f 0 1 (1 + 2 J )e -4π 2 /h 2 J σ β J .
The reason for considering different scale parameters in the construction, is to deal with fat tail, the heuristic being that in the tail we do not require as precise an approximation as in the center. In particular small values of j will be used to estimate the function far off in the tails. To formalize this, we define ζ j := 2 (J-j)(2β/p) , and

A j := [-ζ j , ζ j ],
for all j = 0, . . . J. We also define I J = [-1, 1], and for all j = 0, . . . , J -1 we set I j := A j \A j+1 . Notice that by definition of K j , we can write,

K 0 (χ σ 0 * f 0 )(x) =: k∈Z u 0k ϕ((x -h J σ 0 k)/σ 0 ) K j (χ σ j * ∆ j-1 )(x) =: k∈Z u jk ϕ((x -h J σ j k)/σ j ), ∀j ≥ 1.
To ease notation, we define µ jk := h J σ j k for all j ≥ 0 and all k ∈ Z. In the sequel we shall need the following subset of indexes,

Λ := (j, k) ∈ {0, . . . , J} × Z : |u jk | > σ β J , |µ jk | ≤ ζ j + 2(β + 1) log σ -1 J .
We prove below that we can approximate f 1 by a finite mixture corresponding to retaining only the components associated to indices in Λ and that we can bound the cardinality of Λ by O(J log Jσ

-2β/p J ) To any (j, k) ∈ Λ we associate U j := {σ : σ j ≤ σ ≤ σ j (1 + σ β J )}, V jk := {µ : |µ -µ jk | ≤ σ j σ β J } and W jk := U j × V jk . We denote by M the set of signed measures M on (0, ∞) × R such that |M (W jk ) -u jk | ≤ σ β J for all (j, k) ∈ Λ, and |M |(W c ) ≤ σ β J
, where W c is the relative complement of the union of all W jk for (j, k) ∈ Λ. For any M ∈ M, we write

f M (x) := ϕ((x -µ)/σ) dM (σ, µ).
In proposition 3.6 we control the cardinality of Λ while in proposition 3.8 we control the error between f M and f 1 on the decreasing sequence of intervals [-ζ j , ζ j ]. Proposition 3.7 provides a crude uniform upper bound on f M and f 0 .

Proposition 3.6 -There is a constant C > 0 depending only on f 0 and

Q 0 such that |Λ| ≤ C[σ -(β+1) J ∧ (J log J)σ -2β/p J ] if p ≤ 2β, and |Λ| ≤ C(J log J)σ -1 J if p > 2β.
Proof. First notice that because of propositions 3.9 and 3.10, we always have the bound

4 f 0 1 σ -1 J ≥ 2 f 0 1 J j=0 σ -1 j ≥ J j=0 k∈Z |u jk | ≥ (j,k)∈Λ |u jk | ≥ σ β J |Λ|. (3.19)
If p ≤ 2β, we define B := 2(β + 1) log 2, so that 2(β + 1) log σ -1

J = B √ J. Now consider those indexes j with ζ j ≤ B √ J.
An elementary computation shows that there are at most log J such indexes. Therefore, recalling that there is a separation of h J σ j between two consecutive µ jk and that there are at most J indexes j with

ζ j > B √ J |Λ| J j=0 4ζ j h J σ j + log J × 2B √ J h J σ J ≤ 4h -1 J σ -2β/p J J j=0 2 -j( 2β p -1) + 2B( √ J log J)h -1 J σ -1 J . (3.20) Because h J √ J
1 by definition, and because p ≤ 2β, the result follows from the last equation and equation (3.19). If p > 2β, the reasoning is the same as in the first part, but we can rewrite in this situation the equation (3.20) as

|Λ| ≤ 4h -1 J σ -1 J J j=0 2 (j-J)(1-2β p ) + 2B( √ J log J)h -1 J σ -1 J .
Since p > 2β, the conclusion is immediate.

Proposition 3.7 -For all x ∈ R, all J > 0 large enough and all M ∈ M, it holds

|f M (x) -f 0 (x)| J 3/2 . Proof. Let I ≡ I(x) := {(j, k) ∈ {0, . . . , J} × Z : |x -µ jk | ≤ 2σ j }.
Then the proof is almost identical to proposition 3.4. It suffices to notice that

• |M |(W jk ) ≤ |M (W jk )-u jk |+|u jk |
is always bounded above by a constant, because of the definition of M, of propositions 3.9 and 3.10.

• |x -µ|/σ ≥ (1/4)|x -µ jk |/σ j whenever (σ, µ) ∈ W jk and (j, k) ∈ Λ ∩ I c , as soon as J is large enough. • |I| ≤ 5Jh -1 J for J ≥ 1. Proposition 3.8 -If f 0 ∈ C β , for all J > 0 large enough, all 0 ≤ j ≤ J, all x ∈ [-ζ j , ζ j ] and all M ∈ M, it holds |f M (x) -f 0 (x)| J 3/2 σ β j .
The proof of proposition 3.8 is given in section 3.C.

Proof of theorem 3.1

As mentioned earlier, the proof of theorem 3.1 boils down to verifying conditions (3.13), (3.14) and (3.15) for the three types of priors.

Case of the location mixture

Kullback-Leibler condition for location mixtures

In this Section we verify condition (3.13) in the case of the location mixture prior, using the results of section 3.3.2 By Chebychev inequality, we have

Q 0 [-σ -2β/p , σ 2β/p ] c ≤ σ 2β Q 0 |X| p .
Then by bringing together results from propositions 3.4 and 3.5, we can find a constant C > 0 such that for all

M ∈ M σ |f M,σ (x) -f 0 (x)| 2 dQ 0 (x) ≤ sup |x|>σ -2β/p |f M,σ (x) -f 0 (x)| 2 Q 0 [-σ -2β/p , σ -2β/p ] c + sup |x|≤σ -2β/p |f M,σ (x) -f 0 (x)| 2 ≤ Cσ 2β (log σ -1 ) 2 . By equation (3.7), we have G σ (U σ ) σ -b 3 σ b 4 β exp(-a 3 /σ). Moreover, there is a separation of h σ σ between two consecutive µ k and h σ σ σ, thus all the V k with k ∈ Λ are disjoint. By assumptions on G µ (see equation (3.8)), α k := αG µ (V k ) σ b 5 (β+1) (1 + |µ k |) -b 6 for all k ∈ Λ.
We also define α c := α(V c ). For σ small enough, there is a constant C > 0 not depending on σ such that α c > C . Moreover, since α has finite variation we can assume without loss of generality that C ≤ α c ≤ 1, otherwise we split V c into disjoint parts, each of them having α-measure smaller than one. With

2 n := Cσ 2β (log σ -1 ) 2 , using that Γ(α) ≤ 2α α-1 for α ≤ 1, it follows the lower bound Π(KL(f 0 , n )) ≥ G σ (U σ )Π α (M σ ) σ -b 3 +b 4 β e -a 3 σ -1 σ β 3eΓ(α c ) k∈Λ σ β e -2|u k | 3eΓ(α k ) exp    -K|Λ| log σ -1 -a 3 σ -1 -2 k∈Λ |u k | - k∈Λ log 1 α k    exp    -K|Λ| log σ -1 -Kσ -1 - k∈Λ log 1 α k    , for a generic constant K > 0. From the definition of α k , it holds k∈Λ log 1 α k |Λ| log σ -1 + k∈Λ log (1 + |µ k |) ,
when σ is small enough. Also,

k∈Λ log (1 + |µ k |) = k∈Λ log (1 + |µ k |) 1{|µ k | ≤ 1} + k∈Λ log (1 + |µ k |) 1{|µ k | > 1} ≤ |{k ∈ Λ : |µ k | ≤ 1}| + |Λ| log 2 + k∈Λ log |µ k | ≤ 2h -1 σ σ -1 + 4|Λ| 2β p log σ -1 |Λ| log σ -1 + σ -1
The fourth term in the rhs of the last equation is bounded above by n . Regarding the third term, for any i ∈ L c we have |x j -µ i |/σ > √ 6 log n for all j = 1, . . . , n. Then the third term is bounded by |K|nϕ( √

6 log n) ≤ Hn 2 n n -2 / log n ≤ n . Since we can always choose m ∈ F n, with |u i -u i | ≤ n -3/2 H -1 for all i ∈ I, |µ i -µ i | ≤ n -3/2-1/b 2 for all i ∈ I, and |σ -σ | ≤ n -3/2-1/b 2 , it follows from proposition 3.11 |f (x j ) -m(x j )| ≤ 2 n + i∈I |u i -u i | + i∈I |u i ||ϕ((x j -µ i )/σ) -ϕ((x j -µ i )/σ )| ≤ 2 n + i∈I |u i -u i | + 4 i∈I |u i | |σ i -σ i | σ i ∨ σ i + i∈I |u i | |µ i -µ i | σ i ∨ σ i ≤ 8 n ,
for all j = 1, . . . , n. Therefore d n (f, m) ≤ 8 n , and the claim is proved with δ := 8. To finish the proof, it suffices to compute the cardinality of F n, . A straightforward computation shows that

|R n | ≤ n 5/2+1/b 1 +1/b 2 √ 6 log n ≤ n 4+1/b 1 +1/b 2 for all n ≥ 1, then log N (c 3 n , F n , d n ) ≤ |I| log n n -3/2 × n 4+1/b 1 +1/b 2 + log n 1/b 1 n -3/2-1/b 2 ≤ H 11 2 + 2 b 1 + 2 b 2 n 2 n ,
where the last line holds when n becomes large enough. Then the lemma is proved with

C := (11/2 + 2/b 1 + 2/b 2 )/64.
Lemma 3.5 -Assume that there is n 0 ∈ N, and

0 < γ 1 ≤ γ 2 < 1 such that n -γ 2 /2 ≤ n ≤ n -γ 1 /2 for all n ≥ n 0 . Then Π(F n (H, n ) c ) exp(-H 4 (1 -γ 2 )n 2 n ) for all n ≥ n 0 .
Proof. We use the fact that M ∼ Π α is almost surely purely-atomic (Kingman, 1992)).

Then from the definition of F n it follows

Π(F c n ) ≤ G σ (σ ≤ n -1/b 2 ) + G σ (σ > n 1/b 1 ) + Π α ∞ i=1 |u i | > n + Π α ∞ i=1 |u i | 1{|u i | ≤ n -1 } > n + Π α |{i : |u i | > n -1 }| > Hn 2 n / log n .
We bound each of the term as follows. By assumption

G σ (σ ≤ n -1/b 2 ) e -a 2 n and G σ (σ > n 1/b 1 ) e -a 1 n . Notice that ∞ i=1 |u i | = |M |,
where |M | denote the total variation of the measure M . Since by definition we have

M d = M 1 -M 2 , with M 1 , M 2
independent Gamma random measures with same base measure α(•), it follows that |Q| has the distribution of a Gamma random variable with shape parameter 2α. Then by Markov's inequality,

Π α ∞ i=1 |u i | > n = Π α e 1 2 |M | > e 1 2 n ≤ 2 2α e -1 2 n .
Also, by the superposition theorem (Kingman, 1992, M 3 and M 4 are almost-surely purely atomic, M 3 has only jumps greater than 1/n (almost surely) which number is distributed according to a Poisson distribution with intensity 2αE 1 (n -1 ), where E 1 denotes the exponential integral E 1 function:

E 1 (x) = ∞ x e -t
t dt. Likewise, M 4 has only jumps smaller or equal to 1/n (almost-surely) which number is almost-surely infinite. Recalling that E 1 (x) = γ + log(1/x) + o(1) for x small, it holds 2αγ ≤ 2αE 1 (1/n) ≤ 6α log n ≤ x n for n sufficiently large, with x n := Hn 2 n / log n. Thus using Chernoff's bound on Poisson distribution, we get

Π α |{i : |u i | > n -1 }| > Hn 2 n / log n ≤ e -2αE 1 (1/n) (e2αE 1 (1/n)) xn x xn n ≤ exp - 1 2 x n log x n . But, log x n = log n + log H -2 log -1 n -log log n ≥ (1 -γ 2 ) log n + log H -log log n ≥ 1 2 (1 -γ 2 ) log n for large n. Therefore, as n → ∞ Π α |{i : |u i | > n -1 }| > Hn 2 n / log n ≤ exp - H 4 (1 -γ 2 )n 2 n .
Finally, we use again Markov's inequality to get

Π α ∞ i=1 |u i | 1{|u i | ≤ n -1 } > n = Π α e n n|M4| > e n 2 n ≤ e -n 2 n exp 2α 1/n 0 (e n nx -1)x -1 e -x dx .
But for x ∈ (0, 1/n), we have e n nx -1 ≤ n(e n nδn -1)x, thus the integral in the previous expression is bounded by 2α(e n -1), which is in turn bounded by 2α(e -1) because

n ≤ 1 if n ≥ n 0 .

Case of the location-scale mixture

Kullback-Leibler condition

By Chebychev inequality, we have

Q 0 [-ζ j , ζ j ] c ≤ ζ -p j Q 0 |X| p .
Therefore, bringing together results from propositions 3.7 and 3.8,

|f M (x) -f 0 (x)| 2 dQ 0 (x) = J j=0 I j |f M (x) -f 0 (x)| 2 dQ 0 (x) + A c 0 |f M (x) -f (x)| 2 dQ 0 (x) J 3 J j=0 σ 2β j Q 0 (I j ) + J 3 Q 0 (A c 0 ).
Then we can find a constant

C > 0 such that |f M (x) -f 0 (x)| 2 dQ 0 (x) ≤ CJ 4 σ 2β
J for all M ∈ M and J large enough.

By equation (3.7), we have G σ (U j ) σ -b 3 j σ b 4 β J exp(-a 3 /σ j ) for all j = 0, . . . J. Moreover, there is a separation of h J σ j between two consecutive µ jk and h J σ j σ j , thus all the W jk with (j, k) ∈ Λ are disjoint. By equation (3.8), we have

α jk := αG σ (U j )G µ (V jk ) σ b 5 (β+1)+b 4 β j exp(-a 3 /σ j )(1 + |µ jk |) -b 6
for all (j, k) ∈ Λ. We also define α c := α(W c ). For J large enough, there is a constant C > 0 not depending on J such that α c > C . Moreover, since α has finite variation we can assume without loss of generality that C ≤ α c ≤ 1, otherwise we split W c into disjoint parts, each of them having α-measure smaller than one. With

2 n := CJ 4 σ 2β J , using that Γ(α) ≤ 2α α-1 for α ≤ 1 and M ⊂ KL(f 0 , n ), it follows the lower bound Π(KL(f 0 , n )) ≥ σ β J 3eΓ(α c ) (j,k)∈Λ σ β J e -2|u jk | 3eΓ(α jk ) ≥ σ β J 3eΓ(α c ) (j,k)∈Λ exp -2|u jk | -β log σ -1 J + log 1 6e + (α jk -1) log α jk ≥ exp -KJ|Λ| -2 (j,k)∈Λ |u jk | -(j,k)∈Λ log α -1 jk , (3.21)
for a constant K > 0 depending only on C and β. We now evaluate the sums involved in the rhs of equation (3.21). As before, be have that (j,k)∈Λ |u jk | ≤ 4 f 0 1 σ -1 J (see for instance the proof of proposition 3.8). Act as in section 3.4.1 to find that

(j,k)∈Λ log α -1 jk J|Λ| + J 3/2 σ -1 J + |Λ|σ -1 J .
The term proportional to |Λ|σ -1 J is entirely responsible for the bad rates in location-scale mixtures, and the aim of the hybridation of next section is to get rid of it. For a constant

K > 0, Π(KL(f 0 , n )) ≥ exp -K |Λ|σ -1 J . Then for an appropriate constant C > 0 we can have Π(KL(f 0 , n )) ≥ e -c 2 n 2 n if 2 n = C [n -2β/(3β+2) (log n) t 1 ∧ n -2β/(2β+1+2β/p) (log n) t 2 ], p ≤ 2β, C n -β/(β+1) (log n) t 3 , p > 2β,

Kullback-leibler condition

Let M ≡ M(β, J, f, Λ) be the set of signed measures constructed in section 3.3.3. For any integer J > 0 let Ω J be the event

Ω J := P σ : P σ [2 -j , 2 -j (1 + 2 -Jβ )] ≥ 2 -J ∀ 0 ≤ j ≤ J .
Then with arguments and constant C > 0 from section 3.4.2, letting 2 n := CJ 4 σ 2β J , we have

Π(KL(f 0 , n )) ≥ Π(M) ≥ Π(M | Ω J )Π σ (Ω J ). But by equation (3.11) we have Π σ (Ω J ) exp(-a 6 J b 7 2 J ) and on Ω J it holds α(W jk ) = αP σ (U j )G µ (V jk ) ≥ α2 -J G µ (V jk ) for all (j, k) ∈ Λ. Then act as in equation (3.21) to find a constant K > 0 such that (recalling that σ J = 2 -J ) Π(KL(f 0 , n )) exp -K(J b 7 ∨ J 1/2 )σ -1 J -KJ|Λ| .
Because of proposition 3.6 we can have Π(KL(f

0 , n )) ≥ e -c 2 n 2 n if for an appropriate constant C > 0 2 n = C [n -2β/(3β+1) (log n) 4-6β/(3β+1) ∧ n -p/(p+1) (log n) 4-p/(p+1) ] p ≤ 2β, C n -2β/(2β+1) (log n) 4-2β(4-b 7 ∨3)/(2β+1) p > 2β.

Sieve construction

We use the same sieve F n (H, ) as in equation (3.22). The definition of F n (j, ) is independent of Π thus the conclusion of lemma 3.4 holds for hybrid location-scale mixtures.

It remains to show that Π(F n (H, ) c ) ≤ exp(-2c 2 n 2 n ), which is the object of the next lemma.

Lemma 3.8 -Assume that there is n 0 ∈ N, and

0 < γ 1 ≤ γ 2 < 1 such that n -γ 2 /2 ≤ n ≤ n -γ 1 /2 for all n ≥ n 0 . Then there is a constant a constant γ 2 < γ < 1 such that Π(F n (H, n ) c ) exp(-H 4 (1 -γ)n 2 n ) for all n ≥ n 0 .
Proof. We proceed as in the proof of lemma 3.7. Following the same steps, we deduce that it is sufficient to prove that

Π α ∞ i=1 |u i | 1{σ i > n 1/b 1 } > n e -2c 2 n , Π α ∞ i=1 |u i | 1{σ i ≤ n -1/b 2 } > n e -2c 2 n .
Since the proofs are almost identical for the two previous conditions, we only prove the first and left the second to the reader. Notice that by equation (3.23) we have

Π α ∞ i=1 |u i | 1{σ i > n 1/b 1 } > n P σ ≤ 16α -2 n P σ (σ > n 1/b 1 ) 2 .
Letting Ω := {P σ : P σ (σ > n 1/b 1 ) < exp(-a 1 n/2)}, with a slight abuse of notation, it follows from equation (3.9)

Π α ∞ i=1 |u i | 1{σ i > n 1/b 1 } > n ≤ Π α ∞ i=1 |u i | 1{σ i > n 1/b 1 } > n Ω + Π σ (Ω c ) -2 n exp(-a 1 n) + exp(-a 4 n).

Proof of theorem 3.2

The proof follows the same lines as Ghosal et al. (2007b) with additional cares. The first step consists on rewriting expectation of the posterior distribution as follows. Let (φ n (• | •)) n≥0 be a sequence of test functions such that for n large enough

Q n 0 [P n 0 [φ n (y | x) | x]] N ( /18, F n , d n ) exp - n 2 n 2 , sup {f :dn(f,f 0 )≥17 n/18}∩Fn Q n 0 P n f [1 -φ n (y | x)] | x exp - n 2 n 2 .
The existence of such test functions is standard and follows for instance from Birgé (2006, proposition 4), or Ghosal and van der Vaart (2007, section 7.7). From here, we bound the posterior distribution in a standard fashion,

Q n 0 [P n 0 [Π x ({f : d n (f, f 0 ) > n } | y, x) | x]] ≤ Q n 0 [P n 0 [Π x (F c n | y, x) | x]] + Q n 0 [P n 0 [Π({f : d n (f, f 0 ) > n } ∩ F n | y, x) | x]].
So that,

Q n 0 [P n 0 [Π x ({f : d n (f, f 0 ) > n } | y, x) | x]] ≤ Q n 0 [P n 0 [Π x (F c n | y, x) | x]] + Q n 0 [P n 0 [φ n (y | x)Π x ({f : d n (f, f 0 ) > n } ∩ F n | y, x) | x]] + Q n 0 P n 0 1 -φ n (y | x) Π x ({f : d n (f, f 0 ) > n } ∩ F n | y, x) | x . (3.24)
Now, to any x ∈ R n , we associate the event

E n (x) := y ∈ R n : F n i=1 p f (x i , y i ) p f 0 (x i , y i ) dΠ x (f ) ≥ exp -(1 + 4c 2 ) n 2 n 4 .
(3.25)

3.A.2. Proof of lemma 3.3

We mostly follow the proof of Hangelbroek and Ron (2010, proposition 1). Writing,

K h,σ f σ (x) = h σ k∈Z ϕ x -hσk σ η y -hσk σ f σ (y) dy = h σ k∈Z ϕ x -hσk σ η y -hσk σ f σ (y) dy = h 2π k∈Z ϕ x -hσk σ η(σξ) f σ (ξ)e iξhσk dξ = η(σξ) f σ (ξ) h 2π k∈Z ϕ x -hσk σ e iξhσk dξ.
Then we can invoke the Poisson summation formula (Härdle et al., 1998, theorem 4.1), which is obviously valid for ϕ, and

k∈Z ϕ x -hσk σ e iξhσk = 1 h m∈Z ϕ σξ + 2πm h e i(σξ+ 2πm h )x/σ .
Therefore, recalling that f σ is supported on [-1/σ, 1/σ] and χ equals 1 on [-1, 1],

K h,σ f σ (x) = 1 2π χ(σξ) f σ (ξ) m∈Z ϕ(σξ + 2πm/h) ϕ(σξ) e i(σξ+ 2πm h )x/σ dξ = f σ (x) + 1 2π m∈Z\{0} f σ (ξ) ϕ(σξ + 2πm/h) ϕ(σξ) e i(σξ+ 2πm h )x/σ dξ.
It follows that,

|K h,σ f σ (x) -f σ (x)| ≤ 1 2π f σ 1 sup ξ∈[-1,1] m∈Z\{0} ϕ(ξ + 2πm/h) ϕ(ξ) . Now, f σ 1 ≤ 2σ -1 f σ ∞ ≤ 2σ -1 f σ 1 ≤ 2σ -1 f 1 , which is finite because of lemma 3.2.
Recalling that by assumption ϕ is Gaussian, it follows for all ξ ∈ [-1, 1] and all h ≤ 1,

m∈Z\{0} ϕ(ξ + 2πm/h) ϕ(ξ) ≤ exp - 1 2 (ξ + 2πm/h) 2 + 1 2 ξ 2 ≤ e -1/2 m∈Z\{0} e -4π 2 m 2 /h 2 ≤ 4e -1/2 e -4π 2 /h 2 .
Then the lemma is proved with C := 8e -1/2 .

3.A.3. Some other technical results on K h,σ

Lemma 3.9 -There is a universal constant C > 0 such that for all x ∈ R, all 0 < h ≤ 1 and all σ > 0, k∈Z |η((x -hσk)/σ)| ≤ Ch -1 . Moreover, η ∈ S.

Proof. We first prove that η ∈ S. Obviously ϕ ∈ S, and therefore so is η. Since the Fourier transform and the inverse Fourier transform are continuous mapping of S onto itself, it is immediate that η ∈ S.

We finish the proof by remarking that x → k∈Z |η((x -hσk)/σ)| is periodic with period hσ, hence it suffices to check that it is bounded for

x ∈ [0, hσ]. If x ∈ [0, hσ], then |x -hσk| ≥ |hσk|/2 for any |k| ≥ 2, so that k∈Z |η((x -hσk)/σ)| ≤ 3 sup u∈R |η(u)| + |k|≥2 |η((x -hσk)/σ)| ≤ 3 η 0,0 + η 2,0 |k|≥2 (1 + |hk|/2) -2
≤ 3 η 0,0 + 4 η 2,0 /h, which concludes the proof of the first assertion with C := 3 η 0,0 + 4 η 2,0 , because of the assumption h ≤ 1.

The following Lemma gives some control on the coefficients of f on η. Proposition 3.9 -Let 0 < h ≤ 1 and a k (f ) := (h/σ) η((y -hσk)/σ)f (y) dyx. Then there are universal constants C, C > 0, depending only on ϕ, such that

k∈Z |a k (f )| ≤ C f 1 σ -1 , and for all k ∈ Z, |a k (f )| ≤ C f ∞ .
Proof. For the first assertion of the proposition, we write,

k∈Z |a k (f )| ≤ h σ k∈Z |f (y)||η((y -hσk)/σ)| dy ≤ σ -1 f 1 sup y∈R h k∈Z |η((y -hσk)/σ)|,
and the conclusion follows from lemma 3.9. The proof of the second assertion is simpler. Indeed,

|a k (f )| ≤ h σ |f (y)||η((y -hσk)/σ)| dy ≤ h f ∞ |η(u)| du,
where the last integral is bounded because η ∈ S by lemma 3.9.

3.B. Proof of lemma 3.1

Let x ∈ R n arbitrary, σ > 0 and h σ log σ -1 := 2π √ β + 1. Recall that from lemmas 3.2 and 3.3 we have

K hσ,σ (χ σ * f 0 )-f 0 ∞ σ β , where K hσ,σ (χ σ * f 0 )(z) := k∈Z u k ϕ((z - h σ σk)/σ). Define S n (x) := ∪ n i=1 {z ∈ R : |z -x i | ≤ σ 2(β + 1) log σ -1 } and Λ(x) := {k ∈ Z : |u k | > σ β , h σ σk ∈ S n (x)}.
Also define U σ := {σ : σ ≤ σ ≤ σ(1 + σ β) }, and for all k ∈ Λ(x) define V k := {µ : |µ -h σ σk| ≤ σ β+1 }. We denote by M σ the set of signed measures

M on R such that |M (V k ) -u k | ≤ σ β for all k ∈ Λ(x) and |M |(V c ) ≤ σ β
, where V c is the relative complement of the union of all V k for k ∈ Λ(x). For any M ∈ M σ , we write f M,σ (z) := ϕ((z -µ)/σ) dM (µ). Act as in proposition 3.5 to find that d n (f, f 0 ) ≤ Ch -2 σ σ β for any M ∈ M σ , with a constant C > 0 not depending on x. By construction of S n (x), for all k ∈ Λ(x) there is at least one x i such that |h σ σk -x i | ≤ σ 2(β + 1) log σ -1 . Then for any k ∈ Λ(x), by definition of β+1) .

G x αG x (V k ) ≥ n -1 hσσk+σ β+1 hσσk-σ β+1 g(z -x i ) dz ≥ a 21 n -1 σ a 22 (
Remarking that |Λ(x)| σ -(β+1) independently of x (see proposition 3.3) and letting

n = C h -2
σ σ β we can mimic the steps of section 3.4.1 to find that

Π x (f : d n (f, f 0 ) ≤ s ) exp -C |Λ(x)| log σ -1 -C |Λ(x)| log n exp(-c 2 n 2 n ),
for a constant C > 0 not depending on x and 2 n defined in the lemma.

3.C. Some technical results on the construction of the approximation in the case of location-scale mixtures

Proposition 3.10 -Let f 0 ∈ C β . For any j ≥ 0, we have

|∆ j (x)| ≤ C f 0 C β σ β j , with the same constant C > 0 as in lemma 3.2. Moreover, ∆ j 1 ≤ 2 f 0 1 for all j ≥ 0. Proof. Notice that ∆ j+1 1 ≤ ∆ j 1 + χ σ j+1 * ∆ j ≤ (1 + χ 1 ) ∆ j 1 , by Young's inequality. Since f 0 ∈ L 1 , this implies ∆ j ∈ L 1 for all j ≥ 0. Since ∆ j+1 (ξ) = ∆ j (ξ) - χ σ j+1 (ξ) ∆ j (ξ), we get ∆ j (ξ) = f 0 (ξ) j l=1 (1 -χ σ l (ξ))
, by induction. Because σ j+1 = σ j /2, and by construction of χ σ l we have χ σm (ξ) χ σ l (ξ) = χ σm (ξ) for any m > l, hence the last equation can be rewritten as ∆ j (ξ) = f 0 (ξ)(1 -χ σ j (ξ)). Then we deduce that ∆ j = f 0 -χ σ j * f 0 . By lemma 3.2, this implies that

|∆ j (x)| ≤ C f 0 C β σ β j . From the same estimate, it is clear that ∆ j ≤ f 0 | 1 + χ σ j * f 0 ≤ 2 f 0 1 .

3.C.1. Proof of proposition 3.8

Let define A(β, J) := (2 log |Λ|+2β log σ -1 J ) 1/2 and J ≡ J (x) := {(j, k) ∈ {0, . . . , J}×Z : |x -µ jk | ≤ 4A(β, J)σ j }. For any M ∈ M we can write

f M (x) -f 0 (x) = (j,k)∈Λ∩J W jk ϕ x -µ σ -ϕ x -µ jk σ j dM (σ, µ) + (j,k)∈Λ∩J [M (W jk ) -u jk ] ϕ x -µ jk σ j + (j,k)∈Λ∩J c W jk ϕ x -µ σ dM (σ, µ) - (j,k)∈Λ∩J c u jk ϕ x -µ jk σ j - (j,k) / ∈Λ u jk ϕ x -µ jk σ j + W c ϕ x -µ σ dM (σ, µ) := r 1 (x) + r 2 (x) + r 3 (x) + r 4 (x) + r 5 (x) + r 6 (x).
The proof follows similar steps as the proof of proposition 3.5. From the definition of A(β, J) and proposition 3.6, we deduce that A(β, J) √ J for J large enough. Also, there is a separation of h J σ j between two consecutive µ jk . Then there are no more than 2A(β, J)σ j /(h J σ j ) = 2A(β, J)h -1 J distinct values of µ jk in an interval of length 2A(β, J)σ j . Thus the bound |Λ ∩ J | ≤ 2(J + 1)A(β, J) J 3/2 holds. It follows from proposition 3.11 that

|r 1 (x)| |Λ ∩ J |σ β J J 3/2 σ β J . Obviously, |r 2 (x)| ≤ ϕ ∞ |Λ ∩ J |σ β J J 3/2 σ β J . Whenever (j, k) ∈ Λ ∩ J c and (σ, µ) ∈ W jk , choosing J large enough so that 1/2 ≤ σ j /σ ≤ 2 and |µ -µ jk | ≤ σ j A(β, J)/2, it holds |x -µ| ≥ A(β, J)σ. Therefore, |r 3 (x)| ϕ(A(β, J))|Λ| ≤ σ β J .
With the same reasoning we get |r 4 (x)| f ∞ σ β J . Regarding r 6 , we have the obvious bound |r 6 (x)| ≤ ϕ ∞ σ β J . The r 5 term is more subtle and constitutes the remainder of the proof.

Let Λ c 1 := {(j, k) ∈ {0, . . . , J} × Z :

|u jk | ≤ σ β J } and K j := {k ∈ Z : |µ jk | > ζ j + 2(β + 1) log σ -1 J }. Assuming that x ∈ [-ζ q , ζ
q ] for some 0 ≤ q ≤ J, we can bound r 5 (x) as follows,

|r 5 (x)| ≤ (j,k)∈Λ c 1 |u jk | ϕ x -µ jk σ j + j≤q k∈K j u jk ϕ x -µ jk σ j + j>q k∈K j u jk ϕ x -µ jk σ j , (3.26)
where the third term of the rhs does not exist if q = J. The first term of the rhs of equation (3.26) is bounded by σ β J sup x∈R J j=0 k∈Z ϕ((x -µ jk )/σ), which is in turn bounded by a constant multiple of J 3/2 σ β J (see for instance the proof of lemma 3.9). Because of propositions 3.9 and 3.10, when x ∈ [-ζ q , ζ q ] we always have

j≤q k∈K j |u jk | ϕ x -µ jk σ j ≤ sup j≤q k∈K j ϕ x -µ jk σ j j≤J k∈Z |u jk | ≤ σ β+1 J j≤J 2 f 0 1 σ -1 j ≤ 4 f 0 1 σ β J .
Regarding the second term of the rhs of equation (3.26), we introduce the sets of indexes

L j ≡ L j (x) := {k ∈ K j : |x -µ jk | ≤ σ j 2(β + 1) log σ -1 J }.
Then, we can split again the sum as

j>q k∈K j u jk ϕ x -µ jk σ j = j>q k / ∈L j u jk ϕ x -µ jk σ j + j>q k∈L j u jk ϕ x -µ jk σ j .
With exactly the same reasoning as before, we get that the first sum of the rhs of the last expression is bounded above by 4 f 1 σ β J . Concerning the second term, for any j ≥ 1 we get from propositions 3.9 and 3.10, together with the definition of u jk , that

|u jk | f C β σ β j .
Since there is h J σ j separation between two consecutive µ jk , we deduce that |L j | ≤ 2h -1 J 2(β + 1) log σ -1 J . Therefore, for J large enough and x ∈ [-ζ q , ζ q ] with 0 ≤ q ≤ J,

|r 5 (x)| f 0 1 σ β J + f 0 C β 2(β + 1) log σ -1 J j>q σ β j √ Jσ β q .
The conclusion of the proposition follows by combining all the preceding points.

3.D. Elementary results

Proposition 3.11 -Let ϕ(x) = exp(-x 2 /2). Then, for all µ 1 , µ 2 ∈ R, and all σ 1 , σ

2 > 0 with 1/2 ≤ σ 1 /σ 2 ≤ 2, sup x∈R ϕ x -µ 1 σ 1 -ϕ x -µ 2 σ 2 ≤ 4 |σ 1 -σ 2 | σ 1 ∨ σ 2 + |µ 1 -µ 2 | σ 1 ∨ σ 2 .
Proof. Without loss of generality we can assume that σ 1 ≤ σ 2 . Using the triangle inequality, we first write

ϕ x -µ 1 σ 1 -ϕ x -µ 2 σ 2 ≤ ϕ x -µ 1 σ 1 -ϕ x -µ 2 σ 1 + ϕ x -µ 2 σ 1 -ϕ x -µ 2 σ 2 ≤ sup u∈R ϕ u + µ 1 -µ 2 σ 1 -ϕ(u) + sup u∈R ϕ σ 1 σ 2 u -ϕ(u) .
(3.27)

The first term of the rhs of equation (3.27) is obviously bounded by |µ 1 -µ 2 |/σ 1 . Regarding the second term of the rhs of equation (3.27),

ϕ σ 1 σ 2 u -ϕ(u) ≤ |σ 1 /σ 2 -1| σ 1 σ 2 ∨ 1 2 sup x x 2 ϕ(x),
which terminates the proof.

Proposition 3.12 -Let X ∼ SGa(α, 1), with 0 < α ≤ 1. Then for any x ∈ R and any 0 < δ ≤ 1/2 we have P{|X -x| ≤ δ} ≥ δe -2|x| 3eΓ(α) .

Proof. Assume for instance that x ≥ 0. Recalling that X is distributed as the difference of two independent Ga(α, 1) distributed random variables, it follows

P{|X -x| ≤ δ} ≥ 1 Γ(α) ∞ 0 y α-1 e -y 1 Γ(α) x+y+δ x+y z α-1 e -z dz dy.
Because α ≤ 1, the mapping z → z α-1 e -z is monotonically decreasing on R + , then the last integral in the rhs of the previous equation is lower bounded by δ(x

+ y + δ) α-1 e -(x+y+δ) ≥ δe -2(x+y+δ) . Then P{|X -x| ≤ δ} ≥ δe -2(x+δ) Γ(α) 2 ∞ 0 y α-1 e -3y dy ≥ 3 -α e -2(x+δ) Γ(α) δ ≥ δe -2|x| 3eΓ(α) .
The proof when x < 0 is obvious.

Introduction

Quantum Homodyne Tomography (QHT), is a technique for reconstructing the quantum state of a monochromatic light beam in cavity (Artiles et al., 2005). Unlike classical optics, the predictions of quantum optics are probabilistic so that we cannot in general infer the result of a single measurement, but only the distribution of possible outcomes.

The quantum state of a monochromatic light beam in cavity is a positive, self-adjoint and trace-class operator ρ acting on the Hilbert space L 2 (R). We should here distinguish the pure states which are projection operators onto one-dimensional subspaces of L 2 (R), and mixed-states which are all the other possible states.

Having prepared a quantum system in state ρ, the aim of the physicist is to perform measurement of certain observables. Mathematically speaking, an observable A is a selfadjoint operator on L 2 (R). A measurement is a mapping which assigns to an observable A and a state ρ a probability measure µ A on R; this mapping is given by the so-called Born-von Neumann formula (Hall, 2013).

Two observables of interest in quantum optics correspond to the measurements of the electric field and the magnetic field of a light beam, and are given respectively by the operator Q and P with domains The derivative in the definitions of D(P) and P is understood in the distributional sense. By virtue of the Heisenberg uncertainty principle (Hall, 2013), the observables P and Q cannot be measured simultaneously; that is there is no joint probability distribution associated to the simultaneous measurement of P and Q. Nevertheless, the Wigner density W ρ : R 2 → R, with respect to the Lebesgue measure on R 2 , as defined below, is the closest object to a joint probability density function associated to the joint measurement of P and Q on a system in state ρ. The Wigner distribution satisfies R 2 W ρ = 1, and its marginals on any direction are bona-fide probability density functions. In general, however, W ρ fails to be a proper joint probability density function, as it can take negative values, reflecting the non classicality of the quantum state ρ. For a pure state ρ ψ , ψ ∈ L 2 (R), the Wigner quasi-probability density of ρ ψ is defined as

D(Q) := {ψ ∈ L 2 (R) : x → xψ(x) ∈ L 2 (R)} and D(P) := {ψ ∈ L 2 (R) : x → ψ (x) ∈ L 2 (R
W ψ (x, ω) := R ψ(x + t/2)ψ(x -t/2)e -2πiωt dt, (x, ω) ∈ R 2 . (4.1)
We delay to later the definition of the Wigner distribution for mixed states, which will follow from the definition for pure states in a relatively straightforward fashion. Here we take the opportunity to say that whenever we will be concerned with pure states, we will identify the state ρ ψ to the function ψ ∈ L 2 (R), and talk abusively about the state ψ.

Although we cannot measure simultaneously the observables P and Q, it is possible to measure the quadrature observables, defined as X θ := Q cos θ + P sin θ for all θ ∈ [0, π]. We denote by X ρ θ the random variable whose distribution is the measurement of X θ on the quantum system in state ρ. Assuming that θ is drawn uniformly from [0, π], the joint probability density function (with respect to the Lebesgue measure on R × [0, π]) for (X ρ θ , θ) is given by the Radon transform of the Wigner distribution W ρ , that is

p ρ (x, θ) := 1 π R W ρ (x cos θ -ξ sin θ, x sin θ + ξ cos θ) dξ, (x, θ) ∈ R × [0, π]. (4.2)
For a pure state ψ ∈ L 2 (R), there is a convenient way of rewriting the previous equation, as stated for example in (Markus et al., 2010, equation 4.14),

p ψ (x, θ) =        1 2π| sin θ| R ψ(z) exp πi cot θ 2 z 2 -πi xz sin θ dz 2 θ = 0, θ = π/2, |ψ(x)| 2 /π θ = 0, | ψ(x)| 2 /π θ = π/2, (4.3)
where ψ is the Fourier transform of ψ (according to the convention defined in the next section of the paper). Equation (4.3) emphasizes that for any (x, θ) we indeed have p ψ (x, θ) ≥ 0, a fact that remains true for mixed states, but which is not so clear from the definition of equation (4.2).

Quantum homodyne tomography is an experiment that allow for measuring the quadrature observables X θ for a monochromatic light beam in cavity in state ρ. Here we consider the situation when we perform identical and independent measurements of X θ on n quantum systems in the same state ρ, with θ spread uniformly over [0, π]. Following Butucea et al. (2007), it turns out that a good model for a realistic quantum homodyne tomography must take into account noise on observations.

In practice, the noise is mostly due to the fact that a number of photons fails to be detected. The ability of the detector to detect photons is quantified by a parameter η ∈ [0, 1], called the efficiency of the detector. When η = 0, then the detector fails to detect all photons, whereas η = 1 corresponds to the ideal case where all the photons are detected. In general, it is assumed that η is known ahead of the measurement process, and η is relatively close to one, according to the physicists. Then, from Butucea et al. (2007, section 2.4), a more realistic model for quantum homodyne tomography is to consider that we observe the random variables (given θ)

Y ρ θ = X ρ θ + 1 -η η X vac θ ,
where X θ ∼ p ρ (• | θ), and X vac θ is the random variable whose distribution is the measurement of X θ on the vacuum state and is assumed independent of X ρ θ . Here we adopt the convention that the vacuum state is the projection operator onto x → 2 -1/4 exp(-πx 2 ). It turns out from equations (4.1) and (4.4) that X vac θ has a normal distribution with mean zero and variance 1 1/(4π). This leads to the following efficiency corrected probability density function of observations,

p η ψ (y, θ) := 2 1 -η R p ψ (x, θ) exp - 2πη 1 -η (x -y) 2 dx. (4.4)
To shorten notations, we define

γ := π(1 -η) 2η
, and G γ (x) := π/γ exp -π 2 x 2 /γ , (4.5) so that we have p η ψ (y, θ) = [p ψ (•, θ) * G γ ](y), where * denote the convolution product. To summarize the statistical model we are considering in this paper, we aim at estimating the Wigner density function W ρ , or better directly the state ρ, from n independent and indentically distributed noisy observations (Y 1 , θ 1 ), . . . , (Y n , θ n ) distributed according to the distribution that has the density function of equation (4.4) with respect to the Lebesgue measure on R × [0, π].

The problem of QHT is a statistical nonparametric ill-posed inverse problem that has been relatively well studied from a frequentist point of view in the last few years, and now quite well understood. We mention here only papers with theoretical analysis of the performance of their estimation procedure. We should classify frequentist methods in two categories, depending on whether they are based on estimating the state ρ, or estimating W ρ (although ρ → W ρ is one-to-one, methods based on estimating W ρ don't permit to do the reverse path from W ρ → ρ).

The estimation of the state ρ from QHT measurements has been considered in the ideal situation (η = 1, no noise) by Artiles et al. (2005), while the noisy setting is investigated in Aubry et al. (2008) under Frobenius-norm risk. For smoothness class of realistic states R(C, B, r), an adaptive estimation procedure has been proposed by Alquier et al. (2013) and an upper bound for the Frobenius-norm risk is given. Goodnessof-fit testing is investigated in Méziani (2008).

Regarding frequentist methods for estimating W ρ , the first result goes back to Guţă and Artiles (2007), where sharp minimax results are given over a class of smooth Wigner functions A(β, r = 1, L), under the pointwise risk. The noisy framework has been considered in Butucea et al. (2007); authors obtain the minimax rates of convergence under the pointwise risk and propose an adaptive estimator over the set of parameters β > 0, r ∈ (0, 1) that achieve nearly minimax rates. In the same time [START_REF] Markus | Nonparametric estimation of the purity of a quantum state in quantum homodyne tomography with noisy data[END_REF] explored the estimation of a quadratic functional of the Wigner function, as an estimator of the purity of the state. In, Aubry et al. (2008) an upper bound for the L 2 -norm risk over the class R(C, B, r) is given. More recently, Lounici et al. (2015) established the first sup-norm risk upper bound over A(β, r, L), as well as the first minimax lower bounds for both sup-norm and L 2 -norm risk; they also provide an adaptive estimator that achieve nearly minimax rates for both sup-norm and L 2 -norm risk over A(β, r, L) for all β > 0 and r ∈ (0, 2).

To our knowledge, no Bayesian nonparametric method has been proposed to address the problem of QHT with noisy data, a gap that we try to fill with this paper. In particular, after having introduced preliminary notions in the next section, we propose two families of prior distributions over pure states that can be useful in practice, namely mixtures of coherent-states and random Wilson series. Regarding mixed-states, we will discuss how we can straightforwardly extend the prior distributions over pure states onto prior distributions over mixed states. After presenting simulation results, we will investigate posterior rates of contraction for random Wilson series in the main section of the paper. Rates of contraction, or even consistency, is still challenging for coherent states mixtures, a fact that will be discussed more thoroughly in section 4.5.2.

Preliminaries

Notations

For x, y ∈ R d , xy denote the euclidean inner product of x and y, and x is the euclidean norm of a vector x ∈ R d . For any function f , we denote by f the involution f (x) = f (-x). We use the notation • p for the norm of the spaces L p (R d ).

We use the following convention for the Fourier transform of a function f ∈ L 1 (R d ).

F f (ω) := f (ω) := R d f (x)e -2πixω dx, ∀ω ∈ R d .
Then, whenever f ∈ L 1 (R d ) and F f ∈ L 1 (R d ), the inverse Fourier transform F -1 F f = f is well defined and given by

f (x) = R d f (ω)e 2πiωx dω, ∀x ∈ R d .
Regarding the space L 2 (R d ), we use the convention that the inner product •, • : L 2 (R d )×L 2 (R d ) → C is linear in the first argument and antilinear in the second argument, that is for two functions f, g ∈ L 2 (R d ) we define f, g := R d f (x)g(x) dx, where z is the complex conjugate of z ∈ C. The unit circle of L 2 (R d ) will be denoted by

S 2 (R d ); that is S 2 (R d ) := {f ∈ L 2 (R d ) : f 2 = 1}.
We shall sometimes encounter the Schwartz space S(R d ); that is the space of all infinitely differentiable functions f : R d → R for which |x α D β f (x)| < +∞ for all α, β ∈ N d , with the convention

x α = x α 1 1 . . . x α d d and D β f = ∂ β 1 +•••+β d f /(∂x β 1 1 . . . ∂x β d d )
. Dealing with probability distributions, we consider the Hellinger distance H 2 (P, Q) := 1 2 ( dP/dλ -dQ/dλ) 2 dλ, for any probability measures P, Q absolutely continuous with respect to a common measure λ.

We denote by P ρ , respectively P η ρ , the distributions that admit equation (4.2), respectively equation (4.4), as density with respect to the Lebesgue measure on R × [0, π]. When ρ ≡ ρ ψ denote a pure state, we denote the previous distribution by P ψ and P η ψ , respectively.

Finally, inequalities up to a generic constant are denoted by the symbols and , where a b means a ≤ Cb for a constant C > 0 with no consequence on the result of the proof.

Coherent states

In quantum optics, a coherent state refers to a state of the quantized electromagnetic field that describes a classical kind of behavior.

Let T x f (y) := f (y -x), M ω f (y) = e 2πiωy f (y), denote the translation and modulation operators, respectively, and g a window function with g 2 = 1; most of time g is chosen as g(x) = 2 -1/4 exp(-πx 2 ). Mathematically speaking, coherent states are pure states ρ ψ , that is projection operators, described by a wave-function ψ belonging to

ψ ∈ L 2 (R) : ψ = T x M ω g (x, ω) ∈ R 2 .
Note that the operators T x and M ω are isometric on L p (R d ) and f p = T x M ω f p for any 1 ≤ p ≤ ∞, all f ∈ L p (R d ) and all x, ω ∈ R. Daubechies et al. (1991) proposed simple Wilson bases of exponential decay. They constructed a real-valued function ϕ such that for some a, b > 0,

Wilson bases

|ϕ(x)| e -a|x| , | ϕ(ω)| e -b|ω| ,
and such that the ϕ lm , l ∈ N, m ∈ 1 2 Z defined by 2πlx) if l = 0 and 2m + l is odd, constitute an orthonormal base for L 2 (R). Following Gröchenig (2001, section 8.5), we may rewrite ϕ lm in a convenient form for the sequel, emphasizing the relationship with coherent states,

ϕ lm (x) :=        ϕ(x -2m) if l = 0, √ 2ϕ(x -m) cos(2πlx) if l = 0 and 2m + l is even, √ 2ϕ(x -m) sin(
ϕ lm = c l T m (M l + (-1) 2m+l M -l )ϕ, (l, m) ∈ N × 1 2 Z, (4.6)
where c 0 := 1/2 and c l := 1/ √ 2 for l ≥ 1.

Prior distributions

We recall that a pure state ρ ψ is a projection operator onto a one-dimensional subspace of L 2 (R). Before giving the methodology for estimating general states, we introduce two types of prior distribution over pure-states. More precisely, we first define two probability distributions over S 2 (R), that can be trivially identified with the set of pure-state through the mapping S 2 (R) ψ → ρ ψ ; then we will show how to enlarge these prior distributions to handle mixed states.

The first prior model is based on Gamma mixtures, whereas the second is based on the Wilson base of exponential decay.

Gamma Process mixtures of coherent states

For any finite positive measure α on the measurable space (X, X ), let Π α denote the Gamma process distribution with parameter α; that is, a Q ∼ Π α is a measure on (X, X ) such that for any disjoints B 1 , . . . , B k ∈ X the random variables Q(B 1 ), . . . , Q(B k ) are independent random variables with distributions Ga(α(B i ), 1), i = 1, . . . , k.

We suggest a mixture of coherent states as prior distribution on the wave function ψ. For a Gamma random measure Q on R 2 × [0, 2π], our model may be summarized by the following hierarchical representation. Recall that P η ψ denote the probability distribution having the density of equation (4.4), with ρ = ρ ψ the projection operator onto ψ.

(Y 1 , θ 1 ), . . . , (Y n , θ n ) i.i.d ∼ P η ψ , with ψ = ψ/ ψ 2 ψ(z) = R 2 ×[0,2π] e iφ T x M ω g(z) Q(dxdωdφ) Q ∼ Π α .

Random Wilson series

Let (ϕ lm ) be the orthonormal Wilson base with exponential decay of section 4.2.3. For any positive number Z, let Λ Z be the spherical array

Λ Z := (l, m) ∈ N × 1 2 Z : l 2 + m 2 < Z 2 .
Also define the simplex ∆ Z in the 2 metric as

∆ Z := p = (p lm ) (l,m)∈Λ Z : (l,m)∈Λ Z p 2 lm = 1, p lm ≥ 0 .
We consider the following prior distribution Π on S 2 (R). Let P Z be a distribution over R + and draw Z ∼ P Z . Given Z, draw p from a distribution G( p lm e iζ lm ϕ lm .

Note that (ϕ lm ) is orthonormal, thus ψ 2 2 = (l,m)∈Λ Z p 2 lm = 1 almost-surely, that is ψ ∈ S 2 (R) almost-surely.

Estimation of mixed states

The set of quantum states is a convex set. According to the Hilbert-Schmidt theorem on the canonical decomposition for compact self-adjoint operators, for every quantum state ρ there exists an orthonormal set ( The (α n ) N n=1 are the non-zero eigenvalues of ρ and (ρ ψn ) N n=1 projection operators onto (ψ n ) N n=1 . Thus every mixed state is a convex linear combination of pure states. In particular, for any state ρ we have

ψ n ) N n=1 in L 2 (R) (finite
W ρ (x, ω) = N n=1 α n W ψn (x, ω),
making relatively straightforward the extension of priors over pure states onto priors over general states. In other words, a prior distribution over general states can be constructed as a mixture of pure states by a random probability measure.

Simulations examples

Simulation procedure

We test the Gamma process mixtures of coherent states on two examples of quantum states, corresponding to the Schrödinger cat and 2-photons states, that are respectively described by the wave functions

ψ x 0 cat (x) := exp(-π(x -x 0 ) 2 ) + exp(-π(x + x 0 ) 2 ) 2 1/4 1 + exp(-2πx 2 0 ) , ψ 2 (x) := 2 -1/4 (4πx 2 -1) exp(-πx 2 ).
Using equations (4.1) and (4.2), it is seen that the conditional density on θ ∈ [0, π] corresponding to the measurement of X θ on the systems in states ψ x 0 cat and ψ 2 are respectively given by

p x 0 cat (x | θ) ∝ √ 2e -2π(x-x 0 cos θ) 2 + √ 2e -2π(x+x 0 cos θ) 2 + 2e -2πx 2 0 √ 2e -2πx 2 cos(4πxx 0 sin θ) e -2πx 2 0 sin 2 θ , and, p 2 (x | θ) = 2 -1/2 (4πx 2 -1) 2 e -2πx 2 .
Note that p x 0 cat (• | θ) is not a mixture density, since one term can take negative values. Conditionally on θ drawn uniformly on [0, π], we simulate n = 2000 observations from the Schrödinger cat state with x 0 = 2 using p x 0 cat (• | θ) and the rejection sampling algorithm with candidate distribution 1 2 N (-x 0 cos θ, 1/(4π)) + 1 2 N (x 0 cos θ, 1/(4π)). Similarly, we simulate n = 2000 observations from the 2-photons state using the rejection sampling algorithm with a Laplace candidate distribution. A Gaussian noise is added to observations according to equation (4.4), where we choose η = 0.95, a reasonable efficiency the physicists say.

Simulation results

We use the algorithm of Naulet and Barat (2015) for simulating samples from posterior distributions of Gamma process mixtures. The base measure α on R 2 × [0, 2π] of the mixing Gamma process is taken as the independent product of a normal distribution on R 2 with covariance matrix diag(1/2, 1/2) and the uniform distribution on [0, 2π].

We ran 3000 iterations of the algorithm with p = 50 particles, leading to an acceptance ratio of approximately 60% for the particle moves and the both datasets. All random-walk Metropolis-Hastings steps are Gaussians, with amplitudes chosen to achieve approximately 25% acceptance rates. All the statistics were computed using only the 2000 last samples provided by the algorithm. Compared to other classical methods in this area, our estimate is non linear, preventing easy computations. To our knowledge, however, none of the current approaches can preserve the physical properties of the true Wigner function (non negativity of marginal distributions, bounds) whereas our approach does guarantee preservation of all physical properties.

Rates of contraction for random series priors

In this section, we establish posterior convergence rates in the quantum homodyne tomography problem, for estimating pure states. Unfortunately, to get such result we need a fine control of the L 2 (R) norm of random functions drawn from the prior distribution, which remains challenging for mixtures of coherent states. However, dealing with Wilson bases, the control of the L 2 (R) norm is straightforward and we are able to obtain posterior concentration rates.

Preliminaries on function spaces

To establish posterior concentration rates, we describe suitable classes of functions that can be well approximated by partial sums of Wilson bases elements; these functional classes are called ultra-modulation spaces. To this aim, we need the following ingredients: the short-time Fourier transform (STFT), a class of windows and a class of weights. For a non-zero window function g ∈ L 2 (R), the short-time Fourier transform of a function f ∈ L 2 (R) with respect to the window g is given by

V g f (x, ω) := f, M ω T x g = R f (t)g(t -x)e -2πiωt dt, (x, ω) ∈ R 2 . (4.7)
We also need a class of analyzing windows g with sufficiently good time-frequency localization properties. Following, [START_REF] Cordero | Gelfand-shilov window classes for weighted modulation spaces[END_REF]; [START_REF] Cordero | Localization operators and exponential weights for modulation spaces[END_REF]; [START_REF] Gröchenig | Spaces of test functions via the stft[END_REF], we use the Gelfand-Shilov space S 1 1 (R). For any d ≥ 1, a function

f : R d → C belongs to the Gelfand-Shilov space S 1 1 (R d ) if f ∈ C ∞ (R d
) and there exist real constants h > 0 and k > 0 such that sup

x∈R d |f (x)e h x | < +∞, sup ω∈R d | f (ω)e k ω | < +∞.
Next, for β > 0, g ∈ S 1 1 (R), and r ∈ [0, 1), we consider the exponential weights on R 2 defined by x → exp(β x r ), and we introduce the class of wave-functions

C g (β, r, L) := ψ ∈ S 2 (R) : R 2 |V g ψ(z)| exp(β z r )dz ≤ L .
(4.8)

The class C g (β, r, L) is reminiscent to modulation spaces (Gröchenig, 2001[START_REF] Gröchenig | Weight functions in time-frequency analysis[END_REF]. Note that it would be interesting to consider C g (β, r, L) for r ≥ 1, since most quantum states should fall in these classes. There is, however, at least two limitations for considering r ≥ 1. First, we use repeatedly in the proofs that exp(β x+y r ) ≤ exp(β x r ) exp(β y r ) for r ≤ 1, which is no longer true when r > 1. The previous limitation is indeed not the more serious concerns, since for r > 1 we could use that exp(β x + y r ) ≤ exp(2 r-1 β x r ) exp(2 r-1 β y r ). The more serious problem is that, to our knowledge, there is no Wilson base for L 2 (R) whose elements fall into C g (β, r, L) for r > 1 and β > 0, L > 0. The case r = 1 is more delicate since it depends on the value of β. For sufficiently small β > 0, the results proved in this paper for r < 1 should also hold for r = 1. Let also notice that, there is a fundamental limit on the growth of the weights in the definition of C g (β, r, L), imposed by Hardy's theorem. If r = 2 and β > π/2, the the corresponding classes of smoothness C g (β, r, L) are trivial for any L > 0 [START_REF] Gröchenig | Hardy's theorem and the short-time fourier transform of schwartz functions[END_REF].

A critical point regarding the class C g (β, r, L) is the dependence on g in the definition. We truly want that for two different windows g 0 and g 1 the corresponding smoothness are the same. Fortunately, we have the following theorem, proved in section 4.A.

Theorem 4.1 -Let g, g 0 ∈ S 1 1 (R). For all β, L > 0 and all 0 ≤ r < 1 there is a constant C > 0, depending only on g, g 0 , such that embedding C g (β, r, L) ⊆ C g 0 (β, r, CL) holds.

The STFT and the Wigner transform both aim at having a time-frequency representation of functions in L 2 (R), and are deeply linked to each other. However, contrarily to the Wigner transform, the STFT has the advantage of being a linear operator, which is one reason why we prefer to state the class C g (β, r, L) in term of the STFT instead of the Wigner transform.

Assumptions and results

Before stating the main result of this paper, we need some further assumptions on the random Wilson base series prior, which we state now. To this aim, we need the following definition of the weighted simplex ∆ w Z (β, r, M ). For a constant M > 0, β > 0 and r ∈

[0, 1) let ∆ w Z (β, r, M ) := p ∈ ∆ Z : (l,m)∈Λ Z p lm exp β(l 2 + m 2 ) r/2 < M .
Then, in the sequel, we assume that

• There is a constant a 0 > 0 such that for any sequence (

x lm ) (l,m)∈Λ Z ∈ [0, 2π] |Λ Z | , P ζ (l,m)∈Λ Z |ζ lm -x lm | 2 ≤ t | Z exp -a 0 Z 2 log t -1
, ∀t ∈ (0, 1).

• P Z (Z < +∞) = 1 and there are constants a 1 , a 2 > 0 and b 1 > 2 + r, such that for all k positive integer large enough

P Z (Z = k) exp(-a 1 k b 1 ), P Z (Z > k) exp(-a 2 k b 1 ).
• For any constant C > 0 and any sequence q ∈ ∆ w Z (β, r, C), there is a constant

a 3 > 0 such that the distribution G(• | Z) satisfy, G (l,m)∈Λ Z |p lm -q lm | 2 ≤ t | Z exp -a 3 Z b 1 -r log t -1 , ∀t ∈ (0, 1).
We further assume that there exist constants a 4 ≥ 0, a 5 , c 0 > 0, and b 5 > b 1 /r such that for x > 0 large enough

G p / ∈ ∆ w Z (β, r, c 0 x a 4 ) | Z ≤ x 1/r exp -a 5 x b 5 .
It is not clear whether or not we can find a distribution G for which the above conditions are satisfied simultaneously for all (β, r, L), eventually with constants a 3 , a 4 , a 5 , b 5 depending on (β, r, L). If such distribution exists, then the rates stated below are easily seen to be adaptive on (β, r, L). In section 4.6, we show that for a given (β, r, L) it is easy to construct a distribution G that satisfies the above conditions, with a 4 = 2/r. However, we believe that the proof for adaptive rates must follow a different path, still to be found.

Under the hypothesis above, we will dedicate the rest of the paper to prove the following theorem.

the statistical model described by equation (4.4), with 0 < η < 1 and γ > 0 defined in equation (4.5). Then for any ψ 0 ∈ C g (β, r, L), there is M > 0 such that

P η,n ψ 0 Π( ψ -ψ 0 2 ≥ M n | (Y 1 , θ 1 ), . . . , (Y n , θ n )) → 0, 2 n = (log n) 2a 4 exp -β log n 2γ r/2 + O(1) .
Note that the same result holds with ψ -ψ 0 2 replaced with W ψ -W ψ 0 2 , because the Wigner transform is isometric from L 2 (R) onto L 2 (R 2 ); see for instance Gröchenig (2001, proposition 4.3.2).

The rates of contraction are relatively slow, a fact that is also pointed out in Butucea et al. (2007). Indeed, the rates are faster than (log n) -a but slower than n -a , for all a > 0. The reason for such bad rates of convergence is to be found in the deconvolution of the Gaussian noise. If one does not carry about deconvoluting the noise, then all the steps in the proof of theorem 4.2 can be mimicked to get weaker a result. In particular, we infer from the results of the paper that the posterior distribution should contracts at nearly parametric rates, i.e. at rate n ≈ n -1/2 (log n) t for some t > 0, around balls of the form ψ ∈ S 2 (R) :

R 2 | W ψ (z) -W ψ 0 (z)| 2 G γ ( z ) 2 dz ≤ 2 n , ( 4.9)
whenever ψ 0 ∈ C g (β, r, L) for some β, L > 0 and r ∈ (0, 1). Moreover, we've made many restrictive assumptions on the prior distribution that can be easily released for those interested only in posterior contraction around balls of the form (4.9). A natural question regarding the rates found in theorem 4.2 concerns optimality. We do not know yet the minimax lower bounds over the class C g (β, r, L) for the L 2 risk. However, Butucea et al. (2007); Aubry et al. (2008); Lounici et al. (2015) consider a class A(α, r, L) that resembles to C g (β, r, L). More precisely, they define

A(α, r, L) := W ρ : | W ρ (z)| 2 exp(2α z r )dz ≤ L 2 .
Identifying ρ ψ with ψ, our proposition 4.7 state the embedding C g (β, r, L) ⊆ A(β/2, r, L). Hence C g (β, r, L) is certainly contained in the intersection of a class A(β/2, r, L) with the set of pure states, and it makes sense to compare the rates. To our knowledge, the only minimax lower bound for the quadratic risk known is for the estimation of a state in A(α, r = 2, L), stated in Lounici et al. (2015). For r ∈ (0, 1), however, upper bounds for the quadratic risk over A(β/2, r, L) are established in Aubry et al. (2008), and coincide with the rates found here. Therefore, we believe that the rates we found in this paper are optimal. Let conclude with a few points that are still challenging at this time. First, the rates (or even consistency) for the coherent states mixtures priors appears difficult to establish with the method employed here; the reason comes from the difficulty to control the norm ψ 2 when ψ is a coherent states mixture. Regarding Wilson based priors, we already discussed the lack of adaptivity, which clearly deserved to be dug in a near future. Finally, it would be interesting to consider priors based on Gabor frames expansions, as they are more flexible than Wilson bases, and should be computationally more efficient than coherent states mixtures. However, Gabor frames suffer from the same evil that coherent Throughout the document, we let D β,r n := (log(n)/β) 1/r . Then we introduce the following events, which we'll use several times in the proof of posterior contraction rates. We introduce a new variation around the basic lines of Ghosal et al. (2000); Ghosal and van der Vaart (2007), permitting to slightly weaken the so-called Kullback-Leibler (KL) condition. We show that we can trade the KL condition for a restricted KL condition; that is prior positivity of the sets

B n (δ n ) :=    ψ : En p η ψ 0 log p η ψ 0 p η ψ ≤ δ 2 n , En p η ψ 0 log p η ψ 0 p η ψ 2 ≤ δ 2 n    .
(4.12)

Although looking trivial, this will ease the proof of our main theorem, since the prior positivity of B n (δ n ) is simpler to prove than the classical positivity of KL balls of Ghosal et al. (2000); Ghosal and van der Vaart (2007).

Decay estimates of the true density

It is a classical fact that in Bayesian nonparametrics we often require tails assumptions on the density of observations to be able to state rates of convergence. Here, the density of observations is quite complicated, as being the convolution of a Gaussian noise with the Radon-Wigner transform of ψ. Since the Wigner transform of ψ interpolates ψ and its Fourier transform, we definitively have to take care about fancy tails assumptions on the density that could be non compatible with the requirements of a Wigner transform. Instead, we show that the decay assumptions on the STFT stated in the definition of C g (β, r, L) directly translate onto the tails of the joint density of observations. We have the following theorem, whose proof is given in section 4.B.1.

Lemma 4.1 -For all β, L > 0 and all r ∈ (0, 1) there is a constant C(β, r, η) > 0 such such that

P η ψ (E c n ) ≤ 2πC(β, r, η)L 2 n -2 and P η,n ψ (Ω c n ) ≤ 2πC(β, r, η)L 2 n -1 for all ψ ∈ C g (β, r, L).

Approximation theory

In order to prove the prior positivity of the sets B n (δ n ), we need to construct a family M n of functions in S 2 (R) that approximate well ψ 0 in the L 2 (R) distance. We will show later that the sets B n (δ n ) contains suitable closed balls around ψ 0 in the norm of L 2 (R).

In the sequel, we need to relate the parameters β, r, L to the decay of the coefficients ψ 0 , ϕ lm of ψ 0 ∈ C g (β, r, L) expressed in the Wilson base. Fortunately, Wilson bases are unconditional bases for the ultra-modulation spaces, and C g (β, r, L) is a subset of the ultramodulation space M 1 β,r . It follows the following lemma (Gröchenig, 2001, theorem 12.3.1).

Lemma 4.2 -Let ψ ∈ C g (β, r, L) for some β, L > 0 and 0 ≤ r < 1. Then there is a constant 0 < C(β, r) < +∞ such that

(l,m)∈Λ∞ | ψ, ϕ lm | exp β(l 2 + m 2 ) r/2 ≤ C(β, r)L.
Having characterized the decay of Gabor coefficients for those ψ ∈ C g (β, r, L), we are now in position to construct functions ψ Z which degree of approximation to ψ 0 ∈ C g (β, r, L) is indexed by the value of Z. In view of section 4.2.3, ψ 0 has the formal decomposition ψ 0 = l,m ψ 0 , ϕ lm ϕ lm , with unconditional convergence of the series in L 2 (R). We define ψ Z such that

ψ Z := (l,m)∈Λ Z ψ 0 , ϕ lm ϕ lm .
Since (ϕ lm ) constitutes an orthonormal base for L 2 (R), lemma 4.2 implies that for any β > 0 and r ∈ (0, 1),

ψ 0 -ψ Z 2 2 = (l,m) / ∈Λ Z | ψ 0 , ϕ lm | 2 ≤ exp(-βZ r ) l,m | ψ 0 , ϕ lm | exp β(l 2 + m 2 ) r/2 ≤ C(β, r)L exp (-βZ r ) ,
because on Λ c Z we have l 2 + m 2 ≥ Z 2 and | ψ 0 , ϕ lm | ≤ ψ 0 2 ϕ lm 2 = 1. Note that ψ Z is not necessarily in S 2 (R), that is in general ψ Z 2 = 1, whence it is not a proper wave-function. We now trade ψ Z for a version ψ Z with ψ Z 2 = 1, keeping the same order of approximation. Indeed, let ψ Z := ψ Z / ψ Z 2 , then since ψ 0 2 = 1,

ψ Z -ψ 0 2 ≤ ψ Z -ψ Z 2 + ψ Z -ψ 0 2 ≤ ψ Z 1 - 1 ψ Z 2 + ψ Z -ψ 0 2 ≤ 2 ψ Z -ψ 0 2 ≤ 2 C(β, r)L exp - βZ r 2 . (4.13) A lower bound on Π(B n (δ n ))
The proof of the lemmas and theorem of this section are to be found in sections 4.B.2 and 4.B.3. To prove the Kullback-Leibler condition, we first construct a suitable set M n ⊂ B n (δ n ), and we'll lower bound Π(B n (δ n )) ≥ Π(M n ). Let ψ Z be the function constructed in section 4.7.1 and c lm := ψ Z , ϕ lm , so that ψ Z = (l,m)∈Λ Z c lm ϕ lm . Then, we define the set M n ≡ M n (Z, U ) as follows, and we'll prove that Z, U can be chosen so that M n (Z, U ) ⊂ B n (δ n ).

M n (Z, U ) :=

Conclusion of the proof

Let summarize what we've done so far, and finalize the proof of theorem 4.2. In lemma 4.10 in appendix, we state sufficient conditions to finish the proof of our main theorem; these conditions involve two parts. First, proving that for a suitable sequence δ n → 0 with nδ 2 n → our prior puts enough probability mass on the balls B n (δ n ) and; the construction of tests functions with sufficiently rapidly decreasing type I and type II errors for testing H 0 : ψ = ψ 0 against H 1 : ψ -ψ 0 2 ≥ n , for those ψ in a set F n of prior probability 1 -exp(-6nδ 2 n ). For the prior considered here, we found in theorem 4.3 that δ n must satisfy nδ 2 n ≥ C(log n) b 1 /r for some C > 0, otherwise the so-called Kullback-Leilbler condition is not met. Regarding the construction of tests, this involved to build explicitly the sets F n in section 4.7.2. From that construction and equation (4.15), we deduce that the required test functions exist, if for some constants K 1 , K 2 > 0 and a sequence u n → ∞

δ 2 n ≤ K 1 exp(-2γu 2 n ) 2 n (log n) 2/r , 2 n ≥ K 2 (log n) 2a 4 exp(-βu r n ).
(4.16)

Since we must also have nδ 2 n ≥ C(log n) b 1 /r , we deduce that the sequence (u n ) n≥1 should satisfy, for a suitable constant C > 0, βu r n + 2γu 2 n -2a 5 (log n) s/2 ≤ log C + log n -r -1 (2 + b 1 -2ra 4 ) log log n.

Finally, we can take, where we've used Young's inequality and the first estimate of proposition 4.9. We have by Gröchenig and Zimmermann (2004, corollary 3.10) that V g 0 g ∈ S 1 1 (R 2 ), thus the second integral in the rhs of the last equation is bounded. The conclusion follows from lemma 4.8.

4.B. Proofs of

From the lemmas above the proof of lemma 4.1 is relatively straightforward, we give it here for the sake of completeness.

Proof of lemma 4.1. We begin with the obvious estimate P η,n ψ (Ω c n ) ≤ nP η ψ (E c n ). The proof is finished by noticing that

P η ψ (E c n ) =

4.B.2. Proofs regarding approximation theory

Proof of lemma 4.3. For all ψ ∈ M n (Z, U ) we have the following estimate. Because (ϕ lm ) is an orthonormal base of L 2 (R), Then the conclusion follows using ψ-ψ 0 2 ≤ ψ-ψ Z 2 + ψ Z -ψ 0 2 and equation (4.13). Then for all (y, θ) ∈ E n (i.e. |y| ≤ D β,r n ) it follows from the definition of G γ that p η ψ (y, θ) ≥ G γ (2D β,r n )P ψ (|X| ≤ D β,r n | θ)/(2π). From proposition 4.10 in appendix, the latter implies for n large enough that for all ψ ∈ M n it holds p η ψ (y, θ) ≥ G γ (2D β,r n )/(4π) whenever (y, θ) ∈ E n . Since ψ 0 ∈ C g (β, r, L), which is a subset of the Schwartz space S(R), and since the Radon transform maps S(R) onto a subset of S(R × [0, 2π]) by Helgason (2011, theorem 2.4), we deduce that there is a constant C = C(ψ 0 , η) > 0 such that for all ψ ∈ M n (Z, U ), The proof now follows similar lines as Shen et al. (2013, lemma B2). The function r : (0, ∞) → R defined implicitly by log x = 2(x 1/2 -1) -r(x)(x 1/2 -1) 2 is nonnegative and decreasing. Thus we obtain, (4.17) where the last line follows from Hölder's inequality. Also, proceeding as in the proof of Shen et al. (2013, lemma B2) we find that e -2πij(2x-m-k) V φϕ(2x -m -k, 2ω -l -j) + (-1) 2k+j e 2πij(2x-m-k) V φϕ(2x -m -k, 2ω -l + j) + (-1) 2m+l e -2πij(2x-m-k) V φϕ(2x -m -k, 2ω + l -j) + (-1) 2m+l (-1) 2k+j e 2πij(2x-m-k) V φϕ(2x -m -k, 2ω + l + j) .

  2 ≤ 2 1 - p η ψ 0 p η ψ -2P η ψ 0 (E c n ) + 2 E c n p η ψ 0 p η ψ + r(λ n ) En p η ψ -p η ψ 0 2 ≤ 2H 2 (P η ψ , P η ψ 0 ) (1 + r(λ n )) + 2P η ψ 0 (E c n ) 1/2 P η ψ (E c n ) 1/2 ,
To ease notations, let f (x, ω; l, m, j, k) := e 4πiω(x-m)-2πij(2x-m-k) V φϕ(2x -m -k, 2ω -l -j).

Let Rf (z, θ) denote the Radon transform of f . A straightforward calculus show that F [Rf (•, θ)](u) = f (u cos θ, u sin θ), where f is the Fourier transform with respect to both variables of f , and F the L 1 Fourier operator. Note that,

V φϕ(x, y)e πixy e -2πi(xξ 1 +yξ 2 ) dxdy = R 2 R ϕ(u)ϕ(x -u)e -2πiuy du e πixy e -2πi(xξ 1 +yξ 2 ) dxdy = ϕ(u)e -πiuy ϕ(t)e 2πit(ξ 1 -y/2) dt e -2πiuξ 1 -2πiyξ 2 dy du = 2e 4πiξ 1 ξ 2 ϕ(t) ϕ(t -2ξ 1 )e -4πitξ 2 dt = 2e 4πiξ 1 ξ 2 V ϕ ϕ(2ξ 1 , 2ξ 2 ). Since the Radon transform is a linear map, we deduce that

|p ψ (x, θ)| 8a -1 (l,m)∈Λ Z p lm 2 ≤ 8a -1 |λ Z | ≤ 8a -1 h 2 (log n) 2/r .
Now p η ψ (y, θ) = [p ψ (•, θ) * G γ ](y), so that conclusion of the proposition follows from Young's inequality.

Proof of proposition 4.6. Using the expression of ϕ lm of equation (4.6), we have Using that F [p ψ (•, θ)](ξ) = W ψ (ξ cos θ, ξ sin θ), and performing the suitable change of variables, we arrive at

V g ϕ lm = c l V g (T m M l ϕ) + (-1) 2m+l c l V g (T m M -l ϕ).
R 2 | W ψ (z) -W ψ 0 (z)| 2 | G γ ( z )| 2 dz ≤ 2(C 0 + M n )H 2 (P η ψ , P η ψ 0 ).
Now, using that the Fourier transform is isometric from L 2 (R) onto itself, and that the Wigner transform is isometric from L 2 (R) onto L 2 (R 2 ), by Gröchenig (2001, proposition 4.3.2), we write

ψ -ψ 0 2 2 = R 2 | W ψ (z) -W ψ 0 (z)| 2 dz = { z ≤u} | W ψ (z) -W ψ 0 (z)| 2 dz + { z >u} | W ψ (z) -W ψ 0 (z)| 2 dz ≤ 1 | G γ (u)| 2 R 2 | W ψ (z) -W ψ 0 (z)| 2 | G γ ( z )| 2 dz + { z >u} | W ψ (z) -W ψ 0 (z)| 2 dz.
Under the hypothesis of the lemma, the second term in the rhs of the last equation is bounded by 4R n when n is large, because by proposition 4.7 we have Since G γ (ξ) = exp(-γξ 2 ), it follows,

ψ -ψ 0 2 2 ≤ 1 | G γ (u)| 2 R 2 | W ψ (z) -W ψ 0 (z)| 2 | G γ ( z )| 2 dz + 4R n ≤ 2(C 0 + M n )e

4.C.3. Construction of global test functions

Proof of theorem 4.4. Let N ≡ N ( √ 2δ 2 n , F n , • 2 ) denote the number of balls of radius √ 2δ 2 n and centers in F n , needed to cover F n . Let (B 1 , . . . , B N ) denote the corresponding covering with centers (ψ 1 , . . . , ψ N ). Now let J be the index set of balls B j with ψ jψ 0 2 ≥ n . Using proposition 4.8, for each of these balls B j with j ∈ J, we can build a test function φ n,j satisfying P η,n ψ 0 φ n,j ≤ exp(-6nδ 2 n ), sup

ψ∈B j P η,n ψ (1 -φ n,j ) ≤ exp(-6nδ 2 n ).
Define the test function φ n := max j∈J φ n,j . Then it holds P η,n ψ 0 φ n ≤ j∈J P η,n ψ 0 φ n,j ≤ N exp(-6nδ 2 n ) and P η,n ψ (1 -φ n ) ≤ min j∈J sup ψ ∈B j P η,n ψ (1 -φ n,j ) ≤ exp(-6nδ 2 n ) for any ψ ∈ F n with ψ -ψ 0 2 ≥ n -√ 2δ 2 n (recall that δ n n ), and hence for any ψ ∈ F n with ψ -ψ 0 2 ≥ n . Proof of proposition 4.2. Let Z = KM for K > 0 integer, and (q lm ) (l,m)∈Λ Z ∈ ∆ w Z (β, r, L) be arbitrary. For any (l, m) ∈ Λ Z , and any sequence (p lm ) (l,m)∈Λ Z ∈ ∆ Z , let define the unnormalized coefficients q lm := q lm (n,p)∈I 1 q 2 np , p lm := p lm (n,p)∈I 1 p 2 np , Note that (l,m)∈I 1 q 2 lm = (l,m)∈I 1 p 2 lm = 1. Moreover, we also have

4.D. Proofs for uniform series prior on simplex

(l,m)∈Λ Z p 2 lm =
(l,m)∈Λ Z q 2 lm = 1; it turns out that q lm = q lm (n,p)∈Λ Z q 2 np , p lm = p lm (n,p)∈Λ Z p 2 np .

By the triangle inequality, the two previous expressions of q lm , p lm yield the bound,

(l,m)∈Λ Z |q lm -p lm | 2 ≤ 2 (l,m)∈Λ Z | q lm -p lm | 2 (l,m)∈Λ Z q 2 lm ≤ 2 (l,m)∈Λ Z | q lm -p lm | 2 .
For any k = 1, . . . , K, define t k := (l,m)∈I k q 2 lm and e lm := q lm t -1 k 1((l, m) ∈ I k ). Note that by construction we have t 1 = 1. With obvious definition for θ k and η lm , we have

(l,m)∈Λ Z |p lm -q lm | 2 ≤ 2 K k=1 (l,m)∈I k |θ k η lm -t k e lm | 2 ≤ 4 K k=1 t 2 k (l,m)∈I k |η lm -e lm | 2 + 4 Z k=2 |θ k -t k | 2 .
We can choose M > 0 large enough to have (l,m)∈I 1 q 2 lm ≥ 1/2; it turns out that K k=1 t 2 k ≤ 2. Moreover, with M > 0 chosen as previously we have

t k exp(βk r M r ) = √ 2e βM r (l,m)∈I k q lm exp(β(k -1) r M r ) ≤ √ 2e βM r
(l,m)∈I k q lm exp(β(l 2 + m 2 ) r/2 ) ≤ √ 2Le βM r , thus the coefficients (t k ) K k=1 are in the support of G(• | Z). By independence structure of the prior, and since K k=1 t 2 k ≤ 2, it suffices to prove that for any t > 0,

4.E. Bounding the posterior distribution

We bound the posterior distribution as follows. Let Ω n be the event of equation (4.11).

Then, with the notation Z i := (Y i , θ i ) and Z n = (Z 1 , . . . , Z n ), for any measurable set U n , (4.22) where

P η,n ψ 0 Π(U n | Z n ) = P η,n ψ 0 (Ω n ) [I n 1 + I n 2 + I n 3 ] + P η,n ψ 0 (Ω c n )I n 4 ,
I n 1 := Ωn Π(U n ∩ F c n | z n ) dP η,n ψ 0 (z n | Ω n ), I n 2 := Ωn φ n (z n )Π(U n ∩ F n | z n ) dP η,n ψ 0 (z n | Ω n ), I n 3 := Ωn (1 -φ n (z n ))Π(U n ∩ F n | z n ) dP η,n ψ 0 (z n | Ω n ), I n 4 := Ω c n Π(U n | z n ) dP η,n ψ 0 (z n | Ω c n ).
This decomposition of the expectation for the posterior distribution serves as a basis for the proof of the next lemma.

Lemma 4.10 -Let δ n → 0 with nδ 2 n → ∞. Assume that there are sets F n ⊂ F with Π(F c n ) ≤ e -6nδ 2 n and a sequence of test functions (φ n ) n≥1 , φ n : (R + × [0, 2π]) n → [0, 1], such that P η,n ψ 0 φ n → 0 and sup ψ∈Un∩Fn P η,n ψ (1 -φ n ) ≤ e -6nδ 2 n . Also assume that Π(B n (δ n ))

e -nδ 2 n , where B n (δ n ) are the sets defined in equation (4.12). Then P η,n ψ 0 Π(U n | Z n ) → 0 as n → ∞.

Proof. The proof looks like Ghosal et al. (2000), with careful adaptions. It is obvious that

I n 4 ≤
  dΠ(ψ) Π(B n ) ,
From the definition of B n and because P η ψ 0 (E n ) ≥ 1/2 for n large enough, we get from Chebychev inequality that for those n,

P η,n ψ 0 (A n | Ω n ) ≤ P η,n ψ 0 (C n | Ω n ) ≤ 1 8nδ 2 n . Hence, Ωn∩An (1 -φ n (z n ))Π(U n ∩ F n | z n ) dP η,n ψ 0 (z n | Ω n ) P η,n ψ 0 (A n ) P η,n ψ 0 (Ω n ) ≤ (nδ 2 n ) -1 P η,n ψ 0 (Ω n )
, and, .

Ωn∩A c n (1 -φ n (z n ))Π(U n ∩ F n | z n ) dP η,n ψ 0 (z n | Ω n ) ≤ e
where the third line follows from Fubini's theorem. Combining the last two results yields P η,n ψ 0 (Ω n )I n 3 → 0. The bound on I n 1 follows exactly the same lines as the bound on I n 3 (see also Ghosal et al., 2000).

4.F. Remaining proofs and auxiliary results

Lemma 4.11 -Let ψ, ψ 0 ∈ S 2 (R). Then, H 2 (P η ψ , P η ψ 0 ) ≤ √ 2H(P ψ , P ψ 0 ) ≤ √ 2 ψ -ψ 0 2 . Moreover, we also have that H(P ψ (• | θ), P ψ 0 (• | θ)) ≤ ψ -ψ 0 2 for all θ ∈ [0, π].

Proof. First, we recall that p η ψ (y, θ) = [p ψ (•, θ) * G γ ](y). The same holds for p η ψ 0 . Then using that the square Hellinger distance is bounded by the total variation distance, which is in turn bounded by the Hellinger distance, One can show easily that the same bound holds when θ = 0 or θ = π/2 (although it is even not necessary). The conclusion of the lemma then follows from the definition of the Hellinger distance and the fact that p v is a probability density. The results for conditional densities is immediate from equation (4.24) since p ψ (x | θ) = πp ψ (x, θ) for any ψ ∈ S 2 (R). Dans les modèles de mélanges, le paramètre à inférer depuis les données est une fonction. On définit une distribution a priori sur un espace fonctionnel abstrait au travers d'une intégrale stochastique d'un noyau par rapport à une mesure aléatoire.

Habituellement, les modèles de mélanges sont surtout utilisés dans les problèmes d'estimation de densités de probabilité. Une des contributions de ce manuscrit est d'élargir leur usage aux problèmes de régressions.

Dans ce contexte, on est essentiellement concernés par les problèmes suivants: -Echantillonage de la distribution a posteriori -Propriétés asymptotiques de la distribution a posteriori -Problèmes inverses, et particulièrement l'estimation de la distribution de Wigner à partir de mesurements de Tomographie Quantique Homodyne.

This dissertation deals with Bayesian nonparametric statistics, in particular nonparametric mixture models. The manuscript is divided into a general introduction and three parts on rather different aspects of mixtures approaches (sampling, asymptotic, inverse problem).

In mixture models, the parameter to infer from the data is a function. We set a prior distribution on an abstract space of functions through a stochastic integral of a kernel with respect to a random measure.

Usually, mixture models were used primilary in probability density function estimation problems. One of the contributions of the present manuscript is to use them in regression problems.

In this context, we are essentially concerned with the following problems : -Sampling of the posterior distribution -Asymptotic properties of the posterior distribution -Inverse problems, in particular the estimation of the Wigner distribution from Quantum Homodyne Tomography measurements.

Statistiques bayésiennes non paramétriques, mélanges, vitesses de convergence, échantillonnage, tomographie quantique homodyne Bayesian nonparametric statistics, mixtures, rates of convergence, sampling, quantum homodyne tomography

  b, b =: a a is equal to b by definition , Lesser than up to generic constant, greater than up to generic constant a ∧ b, a ∨ b

  For x ∈ R n , we define x p p,n := n i=1 |x i | p .

  Theorem 1.1 -Bayes' rule. -Assume that {P (n) θ

Definition 1. 4 -

 4 Minimax risk. -The minimax risk over Θ 0 ⊂ Θ associated with a model {P (n) θ : θ ∈ Θ} and with a semimetric d n is R * n := inf θn sup θ∈Θ 0 P

  )}. The operarors Q and P act on D(Q), respectively D(P), as Qψ(x) = xψ(x), and Pψ(x) = -iψ (x).

Figure 2

 2 Figure 2.1: Time evolution of the log-likelihood for different starting point of the Markov Chain, chosen according to the prior distribution, and various parameters of the algorithm. The figure are taken from the test function blip of the section 2.4.1.

  2.3 and 2.4, where

Figure 2 Figure 2

 22 Figure 2.3: Example of simulation results using location-scale mixtures of Gaussians.The root signal-to-noise ratio is equal to 3 for sample size of 1024 design points. The true regression function is represented with dashes, the mean of the sampled posterior distribution in blue and sampled 95% credible bands in pink.

Figure 2

 2 Figure 2.5: Simulation of X-ray computed tomography imaging using symmetric Gamma process location-scale mixture of Gaussians. On the left: the original image. On the right: the reconstructed image from 32768 observations of the Radon transform of the original image in a Gaussian noise.

Lemma 2. 1 -

 1 Let R ⊂ A denote the ring of bounded Borel sets of X . Then the Borel σ-algebra of M(X ) (given the weak-* topology) coincides with the σ-algebra generated by the sets {{µ ∈ M : µ(B) ∈ A} : A ∈ B(R), B ∈ R} and also {{µ ∈ M : µ

  net of the group of d × d orthogonal matrices equipped with spectral norm • , and define

properly f 0

 0 ; otherwise we could have troubles at the boundaries of [-S, S] d . Clearly, without any precaution, h -d K hI * f 0 (x) → f 0 (x)/2 as h → 0 when x belongs to the boundary of [-S, S] d . De Jonge and van Zanten (2010) assume that the covariates are spread onto [a, b] d with a > -S and b < S and extend f 0 by multiplying it by a smooth function that equal 1 on [a, b] d and 0 outside [-S, S] d . Here we assume that the covariates are spread onto [-S, S] d and we use Whitney's extension theorem

  define two sequences of numbers by the following recursion. If |α| = 1 set c α = 0 and d α = -1/α!, and for |α| ≥ 2 define

Proposition 3. 1 -

 1 The inverse-Gaussian distribution with parameters b, a > 0 satisfies equations (3.5) to (3.7) with a1 = b/(2a 2 ), b 1 = 1, a 2 = b/4, b 2 = 1, b 3 = 1, b 4 = 1 and a 3 = b/2.

  For any σ > 0 we use the shortened notation χ σ (ξ) := χ(2σξ). Define η as the function which L 1

  section 2), for any M ∼ Π α we have M d = M 3 + M 4 , where M 3 and M 4 are independent random measures with total variation |M 3 | and |M 4 | having Laplace transforms (for all t ∈ R for which the integrals in the expressions converge) Ee t|M 3 | := exp 2α ∞ 1/n (e tx -1)x -1 e -x dx , Ee t|M 4 | := exp 2α 1/n 0 (e tx -1)x -1 e -x dx .

  )}. The operarors Q and P act on D(Q), respectively D(P), as Qψ(x) = xψ(x), and Pψ(x) = -iψ (x).

  or infinite, in the latter case N = ∞), and α n > 0 such that ρ =

Figure 4 Figure 4

 44 Figure 4.1: Left: Average of Wigner distribution samples from the posterior distribution of the mixture of coherent states prior given 2000 quantum homodyne tomography observations simulated from a Schrödinger cat state. Right: View map of the absolute value of the difference between the posterior mean estimate of the Wigner distribution and the true Wigner distribution.

Figures 4 .

 4 Figures 4.1 and 4.3 represent the average of posterior samples of the Wigner distribution for the Schrödinger cat state, and the 2-photons state, respectively. Because it is hard to distinguish between the posterior mean estimator and the true Wigner distribution, we added to the figures a view map of the absolute value of the difference between the evaluated posterior mean and the true Wigner distribution. Figures 4.2 and 4.4 show the marginals of the posterior mean estimates of Wigner distributions for our two examples. We represented the true marginals in dashed lines, as well as the posterior credible bands provided by the algorithm, which we computed by retaining the 95% samples with the smaller sup-norm distance from the posterior mean estimator of the marginals.Compared to other classical methods in this area, our estimate is non linear, preventing easy computations. To our knowledge, however, none of the current approaches can preserve the physical properties of the true Wigner function (non negativity of marginal distributions, bounds) whereas our approach does guarantee preservation of all physical properties.

E

  n := (y, θ) ∈ R × [0, 2π] : |y| ≤ D β,r n ,(4.10)Ω n := {((y 1 , θ 1 ), . . . , (y n , θ n )) : (y i , θ i ) ∈ E n ∀i = 1, . . . , n}.

ψ

  ∈ S 2 (R) :ψ = (l,m)∈Λ Z p lm e iζ lm ϕ lm , (l,m)∈Λ Z |p lm -|c lm || 2 ≤ U 2 (l,m)∈Λ Z |ζ lm -arg c lm | 2 ≤ U 2

2 (

 2 log n) r/2 ) and the conclusion of the proof follows by equation (4.16). because r < 1 by assumption,R 2 |V g 0 ψ(z)| exp(β z r ) dz ≤ R |V g ψ(z)| * |V g 0 g(z)| exp(β z r ) dz ≤ R 2 R 2 |V g ψ(u) exp(β u r )|V g 0 g(z -u)| exp(β z -u r ) dudz ≤ R 2 |V g ψ(u)| exp(β u r ) du R 2 |V g 0 g(u)| exp(β u r ) du,

  Kullback-Leibler neighborhoods prior mass 4.B.1. Proof of lemma 4.1 To prove lemma 4.1, we need the following intermediate lemmas, relating the smoothness of ψ to the tails of the Wigner density of ψ.

Lemma 4. 7 -

 7 Let ψ ∈ C g (β, r, L) with β, L > 0 and r ∈ (0, 1). Then,R 2 |W ψ (z)| exp(β 2z r )dz ≤ L 2 .Proof. Let ψ(x) = ψ(-x). Then from the definition of V g ψ and W ψ we have that W ψ (x, ω) = 2e 4πiωx V ψψ(2x, 2ω). By lemma 4.6 (with | g, g | = g 2 2 = 1), proposition 4.9, and Young's inequality,|W ψ (z/2)| exp(β z r )dz ≤ 2 (|V g ψ| * |V ψg|)(z) exp(β z r )dz ≤ 2 |V g ψ(u)| exp(β u r )|V ψg(z -u)| exp(β z -u r ) dudz ≤ 2 |V g ψ(z)| exp(β z r )dz × |V ψg(z)| exp(β z r )dz.Moreover, a straightforward computation shows thatV ψg(x, ω) = e -2πiωx V g ψ(x, -ω),which concludes the proof.

Lemma 4. 8 -

 8 Let ψ ∈ C g (β, r, L), with β, L > 0 and r ∈ (0, 1). Then,sup θ R p ψ (x, θ) exp(2β|x| r )dx ≤ L 2 . Proof. From the definition of p ψ , R p ψ (x, θ) e 2β|x| r dx = R 2 W ψ (x cos θ -ξ sin θ, x sin θ + ξ cos θ) e 2β|x| r dξdx.Performing the change of variable (x, ξ) → (x cos θ + ξ sin θ, -x sin θ + ξ cos θ), we arrive atR p ψ (x, θ) e 2β|x| r dx = R 2 W ψ (x, ξ) e 2β|x cos θ+ξ sin θ| r dξdx.But for all r ∈ (0, 1), by the triangle inequality and Hölder's inequality|x cos θ + ξ sin θ| r ≤ (|x cos θ| + |ξ sin θ|) r ≤ (|x| + |ξ|) r ≤ 2 r/2 (x 2 + ξ 2 ) r/2 . Then R p ψ (x, θ) e 2β|x| r dx ≤ R 2 |W ψ (z)| exp (β 2z r ) dz,and the conclusion follows from lemma 4.7.Lemma 4.9 -For all β, L > 0 and r ∈ (0, 1) there is a constantC(β, r, η) > 0 such that if ψ ∈ C g (β, r, L) we have sup θ R p η ψ (y, θ) exp(2β|y| r )dy ≤ C(β, r, η)L 2 .Proof. Using Fubini's theorem twice and the estimate |u + x| r ≤ |u| r + |x| r , p η ψ (y, θ) e 2β|y| r dy = π γ p ψ (x, θ) expπ 2 (x -y) 2 γ dx e 2β|y| r dy = π γ p ψ (x, θ) expπ 2 u 2 γ exp (2β|u + x| r ) dudx ≤ π γ p ψ (x, θ) e 2β|x| r dx expπ 2 u 2 γ + 2β|u| r du.

  θ) e 2β|y| r e -2β|y| r dydθ ≤ n -2 p η ψ (y, θ) e 2β|y| r dydθ ≤ 2πC(β, r, η)L 2 n -2 ,because of lemma 4.9.

  m)∈Λ Z |p lm e iζ lm -c lm | 2 ≤ 2 (l,m)∈Λ Z |p lm -|c lm || 2 + 2 (l,m)∈Λ Z |ζ lm -arg c lm | 2 ≤ 4U 2 .

Proof of lemma 4. 4 .

 4 Recall that p η ψ (y, θ) = [p ψ (•, θ) * G γ ](y).We have the obvious boundp η ψ (y, θ) = +∞ -∞ p ψ (x, θ)G γ (y -x) dx ≥ +D β,r n -D β,r n p ψ (x, θ)G γ (y -x) dx.

  n , ∀(y, θ) ∈ E n .

≤

  H 2 (P η ψ , P η ψ 0 ) 12 + 2r(λ n ) 2 . (4.18)Recalling that T x ϕ(y) = ϕ(y -x) and M ω ϕ(y) = e 2πiωy ϕ(y), it followsR T m M l ϕ(x + t/2)T k M j ϕ(x -t/2)e -2πiωt dt = R e 2πil(x+t/2-m) ϕ(x + t/2 -m)e -2πij(x-t/2-k) ϕ(x -t/2 -k)e -2πiωt dt = 2e 4πiω(x-m)-2πij(2x-m-k) R ϕ(u)ϕ(-u + 2x -m -k)e -2πiu(2ω-l-j) du = 2e 4πiω(x-m)-2πij(2x-m-k) V φϕ(2x -m -k, 2ω -l -j).Thus, we deduce the following expression for the Wigner transform of an arbitrary function ψ ∈ F n .W ψ (x, ω) = (l,m)∈Λ Z (j,k)∈Λ Z p lm p jk e i(ζ lm -ζ jk ) × 2c l c j e 4πiω(x-m)

  u cos θ, u sin θ; l, m, j, k)| = 1 2 |V ϕ ϕ(u cos θ + j -l, u sin θ + m -k)|By Fourier duality, this implies thatsup x |Rf (•; l, m, j, k)(x, θ)| ≤ 1 2 |V ϕ ϕ(u cos θ + j -l, u sin θ + m -k)| duThe function ϕ is in S 1 1 (R) by construction. From Gröchenig and Zimmermann (2004, corollary 3.10) we can then find a constant a > 0 such that it holds|V ϕ ϕ(x, ω)| exp(-a √ x 2 + ω 2 ). Moreover, (u cos θ + j -l) 2 + (u sin θ + m -k) 2 = (u + (j -l) cos θ + (m -k) sin θ) 2 + ((m -k) cos θ -(j -l) sin θ) 2 ≥ (u + (j -l) cos θ + (m -k) sin θ) 2 . l, m, j, k)(x, θ)| exp(-a|u|) du = 2a -1 .

Since

  |V g (T m M l ϕ)(x, ω)| = |V g (x -m, ω -m)|, it follows |V g ϕ lm (x, ω)| ≤ c l |V g ϕ(x -m, ω -l)| + c l |V g ϕ(x -m, ω + l)|.Now pick an arbitraryψ ∈ F n . We have R 2 |V g ψ(z)| exp(β z r ) dz ≤ (l,m)∈Λ Z p lm R 2 |V g ϕ lm (z)| exp(β z r ) dz ≤ (l,m)∈Λ Z p lm c l R 2 |V g ϕ(x -m, ω -l)| exp β(x 2 + ω 2 ) r/2 dxdω + (l,m)∈Λ Z p lm c l R 2 |V g ϕ(x -m, ω + l)| exp β(x 2 + ω 2 ) r/2 dxdω ≤ 2 (l,m)∈Λ Z p lm exp β(l 2 + m 2 ) r/2 R 2 |V g ϕ(z)| exp(β z r ) dz (l,m)∈Λ Z p lm exp β(l 2 + m 2 ) r/2 (log n) a 4 ,where the last line follows fromGröchenig and Zimmermann (2004, corollary 3.10), since both g and ϕ are in S 1 1 (R) and r < 1 by assumption. The previous estimate show that F n ⊂ C g (β, r, L n ) with L n (log n) a 4 . Hence the conclusion follows from proposition 4.7.Recall that F denote the L 1 -Fourier transform operator. Then by Parseval-Plancherel formula we can rewrite|F [p η ψ (•, θ)](ξ) -F [p η ψ 0 (•, θ)](ξ)| 2 dξdθ ≤ 2(C 0 + M n )H 2 (P η ψ , P η ψ 0 ). Recalling that p η ψ (y, θ) = [p ψ (•, θ) * G γ ](y), where F [G γ ] = G γ , it follows F [p η ψ (•, θ)](ξ) = F [p ψ (•, θ)](ξ) G γ (ξ). Therefore, |F [p ψ (•, θ)](ξ) -F [p ψ 0 (•, θ)](ξ)| 2 | G γ (ξ)| 2 dξdθ ≤ 2(C 0 + M n )H 2 (P η ψ , P η ψ 0 ).

| 2 |

 2 W ψ 0 (z)| 2 dz = { z >u} | W ψ 0 (z)| 2 e β z r e -ν z r dz ≤ e -βu r R W ψ 0 (z)| 2 e β z r dz ≤ L 2 e -βu r .

  2γu 2 H 2 (P η ψ , P η ψ 0 ) + 4R n . Consequently, when ψ -ψ 0 2 2 ≥ 8R n we have ψ -ψ 0 2 2 ≤ 4(C 0 + M n )e 2γu 2 H 2 (P η ψ , P η ψ 0 ).

Proof of proposition 4. 1 .

 1 From the definition of of G and Hölder's inequality, forK ≥ 0 integer, Z = KM and (p lm ) (l,m)∈Λ Z in the support of G(• | Z), we get estimate (l,m)∈Λ Z p lm exp(β(l 2 + m 2 ) r/2 ) ≤ K k=1 θ k (l,m)∈I k η lm exp(β(l 2 + m 2 ) r/2 ) ≤ K k=1 θ k |I k | exp(βk r M r ), because K k=1 θ 2 k ≥ θ 2 1 = 1.The conclusion is direct because θ 1 = 1 and θ k is upper bounded by √ 2L exp(-β(k r -1)M r ) for any k = 2, . . . , K.

  m)∈I k |η lm -e lm | 2 ≤ t exp(-cK b 1 -r log t -1 ), (4.20)P K k=2 |θ k -t k | 2 ≤ t exp(-c K b 1 -r log t -1 ), (4.21) for some constants c, c > 0. Equation (4.20) is automatically satisfied by the assumptions on F 1 , F 2 , . . . in the proposition. Equation (4.21) is straightforward from the definition of G(• | Z).

  i , θ i ) p η ψ 0 (y i , θ i ) dΠ(ψ)dP η,n ψ 0 (z n | Ω n ) i , θ i ) p η ψ 0 (y i , θ i ) dP η,n ψ 0 (z n | Ω n )dΠ(ψ) ≤ e 4nδ 2 n Π(U n ∩ F n ) Π(B n ) sup ψ∈Un∩Fn P η,n ψ (1 -φ n ) P η,n ψ 0 (Ω n )

  ψ (•, θ) * G γ ](y) -[p ψ 0 (•, θ) * G γ ](y)| dydθ ≤ G γ 1 |p ψ (x, θ) -p ψ 0 (x, θ)| dxdθ ≤ √ 2H(P ψ , P ψ 0 ),where the second line follows from Young's inequality. Now let θ = 0 and θ = π/2. Using that |x| -|y| = |x -y + y| -|y| ≤ |x -y| for all x, y ∈ C, it holds from equation (4.3) that,p ψ (x, θ) -p ψ 0 (x, θ)On almost recognize the expression of the square-root of a density in the rhs of the last equation. Indeed, it is not because ψ -ψ 0 is not normalized in L 2 . But, lettingψ v := (ψ -ψ 0 )/ ψ -ψ 0 2 , p ψ (x, θ) -p ψ 0 (x, θ) 2 ≤ p v (x, θ) ψ -

Proposition 4. 10 -

 10 There exists n 0 such that for all n ≥ n 0 and all ψ ∈ M n (Z, U ) it holdsP ψ (|X| ≤ D β,r n | θ) ≥ 1/2 for all θ ∈ [0, π].Proof. It suffices to write that,P ψ 0 (|X| ≤ D β,r n | θ) ≤ [-D β,r n ,+D β,r n ] p ψ (x | θ) dx + R |p ψ (x | θ) -p ψ 0 (x | θ)| dx ≤ P ψ (|X| ≤ D β,r n | θ) + √ 2H(P ψ (• | θ), P ψ 0 (• | θ)).By lemma 4.11,√ 2H(P ψ (• | θ), P ψ 0 (• | θ)) ≤ 1/4 for all ψ ∈ M n if n is large enough. Moreover, is n is sufficiently large, we also have P ψ 0 (|X| ≤ D β,r n | θ) ≥ 3/4,concluding the proof. sur les statistiques bayésiennes non paramétriques. La thèse est divisée en une introduction générale et trois parties traitant des aspects relativement différents des approches par mélanges (échantillonage, asymptotique, problème inverse).

Table of contents

 of Résumé . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mean regression with normal errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Multivariate inverse problem example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3 A general result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.4 Supplementary assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

	2.4	Examples of simulations
	2.4.1
	2.5	Rates of convergence
	2.5.1
	2.5.5

•

  Sample from (x 1 , x 2 , . . . )|c 1 , . . . , c n , y 1 , . . . , y n .

	Note that Wolpert et al.'s algorithm serves as a reference to assess the performance
	of our algorithm in chapter 2.

Wolpert's Birth-Death algorithm Wolpert et al.'s algorithm is based on a finite approximation of the Poisson representa- tion of symmetric Gamma process mixtures (see section

  1.2.2). Note that the Poisson representation is not restricted to symmetric Gamma Process mixtures, and so is Wolpert et al.'s algorithm.

Table 2

 2 Mean over 100 runs of RMSE versus acceptance rate in step 3 of the algorithm for some typical test functions. For each signal the number of covariates is set to 128 and the RNSR is equal to 3. credible bands were drawn retaining the 95% samples with the smaller 2 -distance with respect to the posterior mean estimator. Although the algorithm samples an approximated version of the model, it is found that the accuracy of credible bands is quite good since the true regression function almost never comes outside the sampled 95% bands, as it is visible in the example of figs. 2.3 and 2.4. Despite the algorithm efficiency, future work should be done to develop new sampling techniques for regression with mixture models, mainly to improve computation cost.

	tshsine
	RMSE
	ACC-RATE

.1: Summary of root mean squared errors of different algorithms for n = 128 covariates and a root signal to noise ratio of 3. the

Table 2

 2 The root signal-to-noise ratio is equal to 3 for sample size of 1024 design points. The true regression function is represented with dashes, the mean of the sampled posterior distribution in blue and sampled 95% credible bands in pink.

		TI-H	Gibbs	RJMCMC
	Function Symm8 Gauss Symm8 Gauss Symm8
	step	0.0276 0.0268 0.0289 0.0282 0.0300
	wave	0.0088 0.0118 0.0108 0.0133 0.0117
	blip	0.0148 0.0162 0.0172 0.0180 0.0183
	blocks	0.0222 0.0230 0.0241 0.0247 0.0256
	bumps	0.0122 0.0132 0.0182 0.0201 0.0232
	heavisine	0.0154 0.0134 0.0139 0.0147 0.0147
	doppler	0.0180 0.0207 0.0196 0.0261 0.0225
	angles	0.0123 0.0120 0.0123 0.0125 0.0128
	parabolas 0.0135 0.0124 0.0132 0.0147 0.0145
	tshsine	0.0107 0.0109 0.0111 0.0131 0.0120
	spikes	0.0110 0.0075 0.0095 0.0095 0.0103
	corner	0.0077 0.0075 0.0081 0.0095 0.0085

.2: Summary of root mean squared errors of different algorithms for n = 1024 covariates and a root signal to noise ratio of 3. compute credible bands. But, as mentioned in

Antoniadis et al. (

  • We denote by | • | d the standard euclidean norm on R d , and, for any x, y ∈ R d , xy is the standard inner product. For any d×d matrix A with real eigenvalues, we denote λ 1 (A) ≥ • • • ≥ λ d (A) its eigenvalues in decreasing order, A := sup x =0 |Ax| d /|x| d its spectral norm, and A max := max i,j |A ij |, where A ij are the entries of A.

  and hence is determined by its moments (because of proposition 2.10), byBillingsley (2008, Theorem 30.2) the equation (2.15) holds if

  • | Z) over the simplex ∆ Z . Independently of p, draw ζ = (ζ lm ) (l,m)∈Λ Z from a distribution P ζ (• | Z) over [0, 2π] |Λ Z |

	and set
	ψ :=
	(l,m)∈Λ Z

  1 so that P η,n ψ 0 (Ω c n )I n 4 → 0 by lemma 4.1. With the same argument we have thatI n 2 ≤ P η,n ψ 0 (Ω n ) -1 P η,n ψ 0 φ n . Now we bound I n 3 .As usual, recalling that the observations are i.i.d we rewriteΠ(U n ∩ F n | z n ) = (y i , θ i )/p η ψ 0 (y i , θ i ) dΠ(ψ)We lower bound the integral in the denominator of equation (4.23) by integrating on the smaller set B n . Consider the events A By Jensen's inequality, we have the inclusionC n ⊆ A n , thus P η,n ψ 0 (A n | Ω n ) ≤ P η,n ψ 0 (C n | Ω n ).Moreover, using that the observations are independent, and Fubini's theorem, we have Likewise, we can bound the variance with respect to P η,n ψ 0 (• | Ω n ), denoted var for the sake of simplicity; with the same arguments as previously,

		var	n i=1 Bn	log	p η ψ 0 (y i , θ i ) p η ψ (y i , θ i )	dΠ(ψ) Π(B n )
									≤	n ψ 0 (E n ) En Bn P η	log	p η ψ 0 (y, θ) p η ψ (y, θ)	dΠ(ψ) Π(B n )	2	dP η ψ 0 (y, θ)
									≤	n ψ 0 (E n ) Bn P η	 	En	log	p η ψ 0 (y, θ) p η ψ (y, θ)	2	dP η ψ 0 (y, θ)
									Un∩Fn ψ n n i=1 p η i=1 p η ψ (y i , θ i )/p η ψ 0 (y i , θ i ) dΠ(ψ)	.	(4.23)
									Bn	n i=1	p η ψ (y i , θ i ) p η ψ 0 (y i , θ i )	dΠ(ψ) Π(B n )	≤ exp(-4nδ 2 n )
		C n i=1 Bn	log	p η ψ 0 (y i , θ i ) p η ψ (y i , θ i )	dΠ(ψ) Π(B n )	≥ 4nδ 2 n .
	P η,n ψ 0	n i=1 Bn	log	p η ψ 0 (y i , θ i ) p η ψ (y i , θ i )	dΠ(ψ) Π(B n )	| Ω n
				=	1 P η,n ψ 0 (Ω n ) Ωn	n i=1 Bn	log	p η ψ 0 (y i , θ i ) p η ψ (y i , θ i )	dΠ(ψ) Π(B n )	dP η,n ψ 0 ( n j=1 dy j dθ j ∩ Ω n )
				=	nP η ψ 0 (E n ) n-1 P η,n ψ 0 (Ω n )	Bn	En	log	p ψ 0 (y, θ) p ψ (y, θ)	dP η ψ 0 (dydθ)	dΠ(ψ) Π(B n )
				=	n ψ 0 (E n ) Bn En P η	log	p ψ 0 (y, θ) p ψ (y, θ)	dP η ψ 0 (dydθ)	dΠ(ψ) Π(B n )	.

n := ((y 1 , θ 1 ), . . . , (y n , θ n )) : n := ((y 1 , θ 1 ), . . . , (y n , θ n )) :

Whereas approximating a smooth density with a non convex mixture is relatively easy, it is not as simple to achieve a nice order of approximation with convex mixtures.

In the original paper, authors consider that the observations are ( √ ηYi, θi) instead of (Yi, θi) in this manuscript, leading to a slightly different definition for the kernel.

Some readers may have noticed that the variance here is different that inButucea et al. (2007). This comes from a different convention for defining the vacuum state.

Theorem 4.2 -Let β, L > 0 and r ∈ (0, 1). Let Π be the random Wilson series prior satisfying the assumptions above, and (Y 1 , θ 1 ), . . . , (Y n , θ n ) be observations coming from

Proof of theorem 4.1. Using lemma 4.6, we have that |V g 0 ψ| ≤ g -2 2 |V g ψ| * |V g 0 g|. Then,

Remerciements

Acknowledgements

The authors are grateful to Judith Rousseau and Trong Tuong Truong for their helpful support and valuable advice throughout the writing of this article, and also to Pr. Robert L. Wolpert for discussions and RJMCMC source code.

Appendix

2.A. Symmetric Gamma distribution

The symmetric Gamma distribution SGa(a, b), with a, b > 0 is the distribution having Fourier transform t → (1 + t 2 /b 2 ) -a . It is easily seen that if X ∼ Ga(a, b) and Y ∼ Ga(a, b), with X and Y independent, then X -Y has SGa(a, b) distribution.

Proposition 2.10 -Let Z ∼ SGa(a, b). Then for any positive integer n,

Moreover, the distribution SGa(a, b) is determined by its moments (in the sense that SGa(a, b) is the only distribution with this sequence of moments).

Proof. From definition of SGa(a, b), the random variable Z is distributed as X -Y , where X, Y ∼ Ga(a, b) and X, Y are independent. Then it is obvious that all odd moments must vanish. For the even moments, we write,

where the last equality can be obtained after some algebra. To see that SGa(a, b) is determined by its moments, we check that Carleman's criteria applies [START_REF] Gut | Probability: A Graduate Course[END_REF], which is straightforward.

Proposition 2.11 -Let X ∼ SGa(α, η), with 0 < α ≤ 1 and η > 0. Then there is a constant C > 0 such that for any x ∈ R and any 0 < δ ≤ (3 + η) -1 we have P(|X -x| ≤ δ) ≥ Cδ e -(3+η)|x| Γ(α) -1 .

Proof. Assume for instance that x ≥ 0. Recalling that X is distributed as the difference of two independent Ga(α, η) distributed random variables, it follows

x+y+δ x+y z α-1 e -ηz dz dy.

Because α ≤ 1, the mapping z → z α-1 e -ηz is monotonically decreasing on R + , then the last integral in the rhs of the previous equation is lower bounded by δ(x + y + Because |Λ| > σ -1 for σ small enough, it follows from all of the above the existence of a constant K > 0, depending only on f , ϕ and Π, such that Π(KL(f 0 , n )) ≥ exp -K |Λ| log σ -1 .

Then for an appropriate constant C > 0, as a consequence of proposition 3.3, we can have Π(KL(f

Sieve construction for location mixtures

We construct the following sequence of subsets of F, also called a sieve. With the notation

.

The next two lemmas show that F n (H, ) defined as above satisfies all the condition stated in equations (3.14) and (3.15) if H and δ are chosen small enough.

Lemma 3.4 -Let x = (x 1 , . . . , x n ) ∈ R n be arbitrary and d n be the empirical L 2 -distance associated with x. Then for any n -1/2 < n ≤ 1, 0 < H ≤ 1 and n sufficiently large there is a constant

Proof. We write F n ≡ F n (H, n ) to ease notations. The proof is based on arguments from Shen et al. (2013), it uses the fact that the covering number

where t 1 := 4 -8β/(3β + 2) , t 2 := 4 -4β/(2β + 1 + 2β/p) and t 3 := 4 -2β/(β + 1).

Sieve construction

Using the notation f M (x) := ϕ((x -µ)/σ) dM (σ, µ), we construct the following sieve.

Lemma 3.6 -Let x = (x 1 , . . . , x n ) ∈ R n be arbitrary and d n be the empirical L 2 -distance associated with x. Then for any n -1/2 < n ≤ 1, 0 < H ≤ 1 and n sufficiently large there is a constant C > 0 not depending on n such that log

The proof is almost identical to lemma 3.4, with the same constant C > 0.

Lemma 3.7 -Assume that there is n 0 ∈ N, and

Proof. We first write the estimate

The first three terms in the rhs above obeys the same bounds as in the proof of lemma 3.5, using the same arguments. The last two term are bounded using the same trick, thus we simply bound the last term and left the other to the reader. Notice that the random variable

For n large, by assumptions on P σ , it holds α(A n ) n . Then by Chebychev inequality, for n large enough

The conclusion follows from the assumptions on

Hybrid location-scale mixtures

Obviously, given the definition of hybrid mixtures (see section 3.4.3), most of the proof is redundant with the location-scale case, and in the sequel we deal only with the parts that differ.

Consider the first term of the rhs of equation (3.24). We can rewrite,

where the third line follows from Fubini's theorem. The same reasoning applies to the other terms of equation (3.24), using the test functions introduced above and 0 1). But under the condition of the theorem, Ghosal et al. (2007b, Lemma 10) implies that Clearly,

Because χ is C ∞ and compactly supported, for any integer q ≥ 0 we have (iu) q χ(u) = (2π) -1 χ (q) (ξ)e iξu dξ. Clearly χ is Schwartz, hence by Fourier inversion we have that

But, by construction χ(0) = 1, and for any q ≥ 1 we have χ (q) (0) = 0. It follows that χ(u)du = 1, and u q χ(u)du = 0 for any q ≥ 1. Whence, letting m be the largest integer smaller than β, and using Taylor's formula with exact remainder term

states, namely the expansions are not unique and it is hard to control from below the L 2 norm of random Gabor expansions.

Example of priors on the simplex

In this section, we construct a prior on the simplex ∆ Z that satisfy the assumptions of section 4.5.2 for a given (β, r). For all k ≥ 1, and a constant M > 0 to be defined later, we define the sets

We assume without loss of generality that Z = KM for an integer

and draw independently sequences

Now we prove that we can chose reasonably M > 0 and the distributions F 1 , F 2 . . . to met the assumptions of section 4.5.2. The proofs of the next two propositions are to be found in section 4.D.

Proposition 4.1 -There is a constant c 0 > 0 such that for any Z ≥ 0 the sequence (p lm ) (l,m)∈Λ Z belongs to ∆ w Z (β, r, c 0 Z 2 ) with G(• | Z) probability one. Proposition 4.2 -Let M > 0 be large enough, K ≥ 0 integer, and Z = KM . Assume that there is a constant c 0 > 0 and a sequence (d k ) K k=1 such that K k=1 d k ≤ c 0 K, and for any sequence

In the previous proposition, some conditions are required on F 1 , F 2 , . . . ; these conditions are indeed really mild. For instance, it follows from Ghosal et al. (2000, lemma 6.1) that the conclusion of proposition 4.2 is valid if η lm := √ u lm where (u lm ) (l,m)∈I k are drawn from Dirichlet distributions with suitable parameters.

Proof of theorem 4.2

The proof of theorem 4.2 follows the classical approach of Ghosal et al. (2000); Ghosal and van der Vaart (2007) for which the prior mass of Kullback-Leibler type neighborhoods need to be bounded from below and tests constructed. See details in section 4.E.

Lemma 4.3 -For all ψ ∈ M n (Z, U ), it holds with the constant C(β, r) of lemma 4.2,

The fact that M n (Z, U ) is included into a suitable L 2 (R) ball around ψ 0 is not enough to prove the inclusion M n (Z, U ) ⊂ B n (δ n ). The next lemma states sufficient conditions for which the inclusion M n (Z, U ) ⊂ B n (δ n ) actually holds true.

Lemma 4.4 -There are constants

, where C(β, r, η) is the constant of lemma 4.1. Now that we have shown that M n (Z, U ) ⊆ B n (δ n ) for suitable choice of Z and U , it is clear that the prior mass of B n (δ n ) is lower bounded by the prior mass of M n (Z, U ), the one is relatively easy to compute. This statement is made formal in the next theorem.

Construction of tests

The approach for constructing tests is reminiscent to [START_REF] Knapik | A general approach to posterior contraction in nonparametric inverse problems[END_REF], where authors provide a general setup to establish posterior contraction rates in nonparametric inverse problems. We define the following sieve. For positive constants c, h to be determined later, and the constant a 4 > 0 of the assumptions

.

Then, we construct test functions with rapidly decreasing type I and type II errors, for testing the hypothesis H 0 : ψ = ψ 0 against the alternative H 1 : ψ ∈ U n ∩ F n , with U n := {ψ ∈ S 2 (R) : ψ -ψ 0 2 ≥ n }, for a sequence ( n ) n≥0 to be determined later. To this aim, we need the following series of propositions about F n , which are proved in section 4.C.1. 1). Proposition 4.5 -There is a constant M > 0, depending only on ϕ and η, such that for all ψ ∈ F n it holds p η ψ ∞ ≤ M h 2 (log n) 2/r . Proposition 4.6 -For all β > 0 and r ∈ (0, 1) there is a constant R > 0 such that for any

The first step in the tests construction consists on bounding, both from below and from above, the Hellinger distance H 2 (P η ψ , P η ψ 0 ) by a multiple constant of ψ -ψ 0 2 , at least for those ψ 0 ∈ C g (β, r, L) and those ψ ∈ F n . To this aim, we need to estimate the decay of W ψ 0 , stated in the next proposition. The remaining proofs for this section can be found in sections 4.C.2 and 4.C.3. Proposition 4.7 -Let ψ ∈ C g (β, r, L) for some β, L > 0 and r ∈ (0, 1). Then

The practical proposition 4.7 allows to upper bound ψ -ψ 0 2 by H(P η ψ , P η ψ 0 ), provided ψ and ψ 0 are sufficiently separated from each other.

Lemma 4.5 -Let β, L > 0, r ∈ (0, 1), C 0 := p η ψ 0 ∞ , M, R > 0 be the constants of propositions 4.5 and 4.6, and assume n large enough. Then for all u > 0, all ψ ∈ F n and all

From the last lemma, we are in position to construct test functions with rapidly decreasing type I and type II error for testing

where (u n ) n≥0 is an increasing sequence of positive numbers to be determined later and M, R > 0 the constants of propositions 4.5 and 4.6.

Proposition 4.8 -Let δ n , n be as in equation (4.15). Then there exist test functions

with type I and type II errors satisfying

Proof. By lemma 4.5, we deduce that The small balls estimate of proposition 4.8 allows to build the desired test functions, using the classical approach of the covering of F n with balls of radius √ 2δ 2 n in the L 2 (R) norm (Ghosal et al., 2000).

Theorem 4.4 -Assume that ψ 0 ∈ C g (β, r, L) for β, L > 0 and r ∈ (0, 1), and let n , δ n be as in equation

Then there exist test functions (φ n ) n≥0 such that We need some subsidiaries results to prove the theorem 4.1.

and sup

Proposition 4.9 -For all β > 0, all 0 ≤ r ≤ 1 and all x, y ∈ R 2 , it holds exp(β x + y r ) ≤ exp(β x r ) exp(β y r ).

Proof. This follows from the trivial estimate

The next lemma is about the change of window in the STFT; its proof is given for arbitrary g ∈ S(R) and ψ ∈ S (R) in Gröchenig (2001, lemma 11.3.3). The proof is identical when g, ψ ∈ L 2 (R), since it essentially rely on a duality argument. Note, however, that the class of windows and functions that we are considering are subset of S(R).

Proof. From Gröchenig (2001, corollary 3.2.3), for those g, h ∈ L 2 (R) with h, g = 0, we have the inversion formula ψ = h, g -1 V g ψ(x, ω) M ω T x h dωdx for all ψ ∈ L 2 . Applying V g 0 both sides

The conclusion follows because

Finally, we have the sufficient material to establish the independence of the class C g (β, r, L) with respect to the choice of the window function g, as soon as g is suitably well behaved.

Note that r(x) ≤ log x -1 for x small enough, and by lemma 4.1,

Then we deduce from equations (4.17) to (4.19) and lemma 4.11 that for n large enough, provided δ 2 n ≥ 4 2πC(β, r, η)Ln -1 ,

Then the conclusion follows from lemma 4.3.

4.B.3. Proof of the lower bound

Proof of theorem 4.3. Let C 1 , C 2 > 0 be the constants of lemma 4.4, and let U n = C 1 (log n) -4/r δ 2 n and Z n be the smaller integer larger than

Note that by lemma 4.2 the sequence (|c lm |) (l,m)∈λ Z is in ∆ w Z (β, r, C(β, r)L). Hence, using the assumptions of section 4.5.2, we have for n large enough

We deduce from the above the existence of a constant K > 0 not depending on n, such that for n large enough,

Then the conclusion of the theorem follows since we assume nδ 2 n = C(log n) b 1 /r for a suitable constant C > 0.

4.C. Proofs of tests construction

4.C.1. Proofs regarding the sieve

Proof of proposition 4.3. Let Z n be the smaller integer larger than h(log n) 1/r . Clearly ψ ∼ Π is almost-surely in S 2 (R). Then if c > 0 is large enough we have the bound

which is trivially smaller than a multiple constant of exp(-6nδ 2 n ) when h is as large as in the proposition, and because b 5 > b 1 /r by assumption. (Shen et al., 2013). We compute the cardinality of such √ 2δ 2 n -net as follows. Let Z n := h(log n) 1/r , P be a δ 2 n -net over the simplex ∆ Zn in the 2 distance, and let O be a δ 2 n -net over [0, 2π] in the euclidean distance. Then define

For all ψ ∈ F n we have ψ = (l,m)∈Λ Zn q lm e iζ lm ϕ lm , with q lm = p lm for those (l, m) ∈ Λ Z , Z ≤ Z n , and q lm = 0 otherwise. Since (ϕ lm ) is an orthonormal base of L 2 (R), we have (l,m)∈Λ Zn q 2 lm = 1, and we can find a function N n ψ = (l,m)∈Λ Zn q lm e iζ lm ϕ lm such that (l,m)∈Λ Zn |q lm -q lm | 2 ≤ δ 4 n , and |ζ lm -ζ lm | ≤ δ 2 n for all (l, m) ∈ Λ Zn . Using standard arguments, we have

, for a constant C > 0. Clearly, the cardinality of a √ 2δ 2 n -net over F n in the • 2 distance satisfy the same bound, eventually for a different constant C. Therefore, for a suitable constant K > 0, when n is large enough.

The conclusion follows because b 1 > 2 + r.

Proof of proposition 4.5. The bound is obvious for those ψ ∈ F n with Z = 0. For Z ≥ 1, we have from the definition of the Wigner transform (equation (4.1)), for an arbitrary

Using the expression of ϕ lm from equation (4.6), it follows

4.C.2. Proofs of norm equivalence

Proof of proposition 4.7. Recall that F denote the L 1 Fourier transfom operator. By definition of W ψ , it holds W ψ (u 1 , u 2 ) = F [ψ(u 1 + •/2)ψ(u 1 -•/2)](u 2 ). Clearly if ψ ∈ C g (β, r, L) then W ψ ∈ L 1 (R 2 ) by lemma 4.7. Moreover, for all u 1 ∈ R the mapping t → ψ(u 1 + t/2)ψ(u 1 -t/2) is in L 1 (R) because of Cauchy-Schwarz inequality and ψ ∈ L 2 (R). Then by Fourier inversion, we get W ψ (u 1 , u 2 )e -2πiu 2 (-ξ 2 ) du 2 = ψ(u 1 + ξ 2 /2)ψ(u 1 -ξ 2 /2).

Taking the Fourier transform with respect to u 1 yields W ψ (u 1 , u 2 )e -2πi(u 1 ξ 1 +u 2 ξ 2 ) du 1 du 2 = ψ(u 1 -ξ 2 /2)ψ(u 1 + ξ 2 /2)e -2πiu 1 ξ 1 du 1 = e -πiξ 1 ξ 2 ψ(t)ψ(t + ξ 2 )e -2πiξ 1 t dt.

Hence we proved that W ψ (ξ 1 , ξ 2 ) = e -πiξ 1 ξ 2 V ψ ψ(-ξ 2 , ξ 1 ), at least when ψ ∈ C g (β, r, L). By lemma 4.6, |V ψ ψ(-ξ 2 , ξ 1 )| ≤ (|V g ψ| * |V ψ g|)(-ξ 2 , ξ 1 ) since g 2 = 1. Note that, by proposition 4.9 we have exp(β(ξ 2 1 + ξ 2 2 ) r/2 ) ≤ exp(β((-ξ 2 -u 1 ) 2 + (ξ 1 -u 2 ) 2 ) r/2 ) exp(β(u 2 1 + u 2 2 ) r/2 ). Taking the integral both sides, under the assumptions of the lemma it comes |p η ψ (y, θ) -p η ψ 0 (y, θ)| 2 dydθ ≤ 2(C 0 + M n )H 2 (P η ψ , P η ψ 0 ).