Pierre Alquier

Karim Lounici

PAC-BAYESIAN BOUNDS FOR SPARSE REGRESSION ESTIMATION WITH EXPONENTIAL WEIGHTS

Keywords: MSC 2000 subject classification: Primary: 62J07; Secondary: 62J05, 62G08, 62F15, 62B10, 68T05 Sparsity Oracle Inequality, High-dimensional Regression, Exponential Weights, PAC-Bayesian, RJMCMC

 but with improved statistical performances. Our main theoretical result is a sparsity oracle inequality in probability for the true excess risk for a version of exponential weight estimator. We also propose a MCMC method to compute our estimator for reasonably large values of p.

Introduction

We observe n independent pairs (X 1 , Y 1 ), ..., (X n , Y n ) ∈ X × R (where X is any measurable set) such that (1.1)

Y i = f (X i ) + W i , 1 i n,
where f : X → R is the unknown regression function and the noise variables W 1 , . . . , W n are independent of the design (X 1 , . . . , X n ) and such that EW i = 0 and EW 2 i σ 2 for some known σ 2 > 0 and any 1 i n. The distribution of the sample is denoted by P, the corresponding expectation is denoted by E. For any function g : X → R define g n = n i=1 g(X i ) 2 /n 1/2 and g = E g 2 n 1/2 .

Let F = {φ 1 , . . . , φ p } be a set-called dictionary-of functions φ j : X → R such that φ j = 1 for any j (this assumption can be relaxed). For any θ ∈ R p define f θ = p j=1 θ j φ j and the risk

R(θ) = E 1 n n i=1 Y i -f θ (X i ) 2 ,
where {(X 1 , Y 1 ), . . . , (X n , Y n )} is an independent replication of the sample. Let us choose θ ∈ arg min θ∈R p R(θ). Note that the minimum may not be unique but since we consider in this paper the prediction problem, we do not need to deal with the identifiability question.

It is a known fact that the least-square estimator θLSE n ∈ arg min θ∈Θ r(θ) performs poorly in high-dimension p > n. Indeed, consider for instance the deterministic design case with i.i.d. noise variables N (0, σ 2 ) and a full-rank design matrix, then θLSE satisfies

E f θLSE n -f 2 n -f θ -f 2 n = σ 2 .
In the same context, assume now there exists a vector θ ∈ arg min θR p R(θ) with a number of nonzero coordinates p 0 ≤ n. If the indices of these coordinates are known, then we can construct an estimator θ0 n such that

E f θ0 n -f 2 n -f θ -f 2 n = σ 2 p 0 n .
The estimator θ0 n is called oracle estimator since the set of indices of the nonzero coordinates of θ is unknown in practice. The issue is now to build an estimator, when the set of nonzero coordinates of θ is unknown, with statistical performances close to that of the oracle estimator θ0 n . A possible approach is to consider solutions of penalized empirical risk minimization problems:

θpen ∈ arg min θ∈Θ 1 n n i=1 Y i -f θ (X i ) 2 + pen(θ) ,
where the penalization pen(θ) is proportional to the number of nonzero components of θ as for instance C p , AIC and BIC criteria [START_REF] Mallows | Some comments on cp[END_REF][START_REF] Akaike | Information theory and an extension of the maximum likelihood principle[END_REF][START_REF] Schwarz | Estimating the dimension of a model[END_REF]. Bunea, Tsybakov and Wegkamp [START_REF] Bunea | Aggregation for Gaussian regression[END_REF] established for the BIC estimator θBIC n the following non-asymptotic sparsity oracle inequality. For any > 0 there exists a constant C( ) > 0 such that for any p 2, n 1 we have

E f θBIC n -f 2 n (1 + ) f θ -f 2 n + C( )σ 2 p 0 n log ep p 0 ∨ 1 .
Despite good statistical properties, these estimators can only be computed in practice for p of the order at most a few tens since they are solutions of non-convex optimization problems.

Considering convex penalty function leads to computationally feasible optimization problems. A popular example of convex optimization problem is the Lasso estimator (cf. Frank and Friedman [START_REF] Frank | A statistical view on some chemometrics regression tools[END_REF], Tibshirani [START_REF] Tibshirani | Regression shrinkage and selection via the lasso[END_REF], and the parallel work of Chen et al on basis pursuit [START_REF] Chen | Atomic decomposition by basis pursuit[END_REF]) with the penalty term pen(θ) = λ|θ| 1 , where λ > 0 is some regularization parameter and, for any integer d ≥ 2, real q > 0 and vector z ∈ R d we define |z| q = ( d j=1 |θ q j |) 1/q and |z| ∞ = max 1≤j≤d |θ j |. Several algorithms allow to compute the LASSO for very large p, one of the most popular is known as LARS, introduced by Efron et al [START_REF] Efron | Least angle regression[END_REF]. However, the Lasso estimator requires strong assumptions on the matrix A = (φ j (X i )) 1 i n,1 j p to establish fast rates of convergence results. Bunea, Tsybakov and Wegkamp [START_REF] Bunea | Sparsity oracle inequalities for the lasso[END_REF] assume a mutual coherence condition on the dictionary. Bickel, Ritov and Tsybakov [START_REF] Bickel | Simultaneous analysis of lasso and dantzig selector[END_REF] and Koltchinskii [START_REF] Koltchinskii | Sparsity in empirical risk minimization[END_REF] established sparsity oracle inequalities for the Lasso under a restricted eigenvalue condition. Candès and Tao [START_REF] Candes | The dantzig selector: statistical estimation when p is much larger than n[END_REF] proposed the Dantzig Selector which is related to the Lasso estimator and suffers from the same restrictions. See for example Bickel, Ritov and Tsybakov [START_REF] Bickel | Simultaneous analysis of lasso and dantzig selector[END_REF] for more details. Several alternative penalties were recently considered. Zou [START_REF] Zou | The adaptive lasso and its oracle properties[END_REF] proposed the adaptive LASSO which is the solution of a penalized empirical risk minimization problem with the penalty pen(θ) = λ p j=1

1 | ŵj | |θ j |
where ŵ is an a priori estimator. Zou and Hastie [START_REF] Zou | Regularization and variable selection via the elastic net[END_REF] proposed the elastic net with the penalty pen(θ) = λ 1 |θ| 1 + λ 2 |θ| 2 2 , λ 1 , λ 2 > 0. Meinshausen and Bühlmann [START_REF] Meinshausen | Stability selection[END_REF] and Bach [START_REF] Bach | Model-consistent sparse estimation through the bootstrap[END_REF] considered bootstrapped LASSO. See also Ghosh [START_REF] Ghosh | Adaptive elastic net: an improvement of elastic net to achieve oracle properties[END_REF] or Cai, Xu and Zhang [START_REF] Cai | On recovery of sparse signals via 1 minimization[END_REF] for more alternatives to the LASSO. All these methods were motivated by their superior performances over the LASSO either from the theoretical or the practical point of view. However, strong assumptions on the design are still required to establish the statistical properties of these methods (when such results exist). A recent paper by van de Geer and Bühlmann [START_REF] Van De Geer | On the conditions used to prove oracle results for the lasso[END_REF] provides a complete survey and comparison of all these assumptions.

Simultaneously, the PAC-Bayesian approach for regression estimation was developed by Audibert [START_REF] Audibert | Aggregated estimators and empirical complexity for least square regression[END_REF][START_REF] Audibert | PAC-Bayesian Statistical Learning Theory[END_REF] and Alquier [START_REF] Alquier | Transductive and Inductive Adaptative Inference for Regression and Density Estimation[END_REF][START_REF] Alquier | Pac-bayesian bounds for randomized empirical risk minimizers[END_REF], based on previous works in the classification context by Catoni [START_REF] Catoni | A pac-bayesian approach to adaptative classification[END_REF][START_REF] Catoni | Statistical Learning Theory and Stochastic Optimization[END_REF][START_REF] Catoni | PAC-Bayesian Supervised Classification (The Thermodynamics of Statistical Learning)[END_REF], Mc Allester [START_REF] Mcallester | Some pac-bayesian theorems[END_REF], Shawe-Taylor and Williamson [START_REF] Shawe-Taylor | A pac analysis of a bayes estimator[END_REF], see also Zhang [START_REF] Zhang | From epsilon-entropy to kl-entropy: analysis of minimum information complexity density estimation[END_REF] in the context of density estimation. This framework is very well adapted for studying the excess risk R(•) -R(θ) in the regression context since it requires very weak conditions on the dictionary. However, the methods of these papers are not computationally feasible when p becomes large. Dalalyan and Tsybakov [19,[START_REF] Dalalyan | Pac-bayesian bounds for the expected error of aggregation by exponential weights[END_REF][START_REF] Dalalyan | Mirror averaging with sparsity priors[END_REF][START_REF] Dalalyan | Sparse regression learning by aggregation and langevin monte-carlo[END_REF] propose an exponential weights procedure related to the PAC-Bayesian approach with good statistical and computational performances. However they consider deterministic design, establishing their statistical result only for the empirical excess risk instead of the true excess risk R(•) -R(θ).

In this paper, we propose to study two exponential weights estimation procedures. The first one is an exponential weights combination of the least squares estimators in all the possible sub-models. This estimator was initially proposed by Leung and Barron [START_REF] Leung | Information theory and mixing least-squares regressions[END_REF] in the deterministic design setting. Note that in the literature on aggregation, the elements of the dictionary are often preliminary arbitrary estimators computed from a frozen fraction of the initial sample so that these estimators are considered as deterministic functions, the aggregate is then computed using this dictionary and the remaining data. This scheme is referred to as 'data splitting'. See for instance Dalalyan and Tsybakov [START_REF] Dalalyan | Pac-bayesian bounds for the expected error of aggregation by exponential weights[END_REF][START_REF] Dalalyan | Mirror averaging with sparsity priors[END_REF] and Yang [START_REF] Yang | Aggregating regression procedures to improve performance[END_REF]. Leung and Barron [START_REF] Leung | Information theory and mixing least-squares regressions[END_REF] proved that data splitting is not necessary in order to aggregate least squares estimators and raised the question of computation of this estimator in high dimension. In this paper we explicit the oracle inequality satisfied by this estimator in the high-dimensional case and tackle the computational question. For the second procedure, the design is assumed to be random. We use the PAC-Bayesian techniques of Catoni [START_REF] Catoni | PAC-Bayesian Supervised Classification (The Thermodynamics of Statistical Learning)[END_REF] to build an estimator satisfying a sparsity oracle inequality for the true excess risk. Then we propose computationally efficient Monte Carlo algorithms to compute both estimators. Our algorithms are inspired from the computational Bayesian theory, see the monograph of Marin and Robert [START_REF] Marin | Bayesian Core: A practical approach to computational Bayesian analysis[END_REF] for an introduction to Monte Carlo algorithms in Bayesian theory. More specifically, the Bayesian point of view for the variable selection problem was considered in several papers: George [START_REF] George | The variable selection problem[END_REF], George and McCulloch [START_REF] George | Approaches for bayesian model selection[END_REF], West [START_REF] West | Bayesian factors in the "large p, small n" paradigm[END_REF], Jiang [START_REF] Jiang | Bayesian variable selection for high dimensionnal generalized linear models: Convergence rate of the fitted density[END_REF], Cui and George [START_REF] Cui | Empirical bayes vs. fully bayes variable selection[END_REF], Bogdan et al [START_REF] Bogdan | Optimal rules for multiple testing and sparse multiple regression[END_REF], Liang et al [START_REF] Liang | Mixture of g-priors for bayesian variable selection[END_REF], Scott and Berger [START_REF] Scott | Bayes and empirical-bayes multiplicity adjustement in the variable-selection problem[END_REF] among others. See in particular [START_REF] George | Approaches for bayesian model selection[END_REF][START_REF] Nott | Sampling schemes for bayesian variable selection in geleralized linear models[END_REF] for the algorithmic aspects of Monte Carlo techniques. In this paper, we use Hastings Metropolis algorithm to compute our first estimator and we implement a version of the RJMCMC ("Reversible Jump Markov Chain Monte Carlo") method proposed by Green [START_REF] Green | Reversible jump markov chain monte carlo computation and bayesian model determination[END_REF] to compute our second estimator. Note that in a work parallel to ours, Rigollet and Tsybakov [START_REF] Rigollet | Exponential screening and optimal rates of sparse estimation[END_REF] consider exponentially weighted aggregates with discrete priors and suggest another version of the Metropolis-Hastings algorithm to compute their estimator.

The paper is organized as follows. In Section 2 we define a general aggregation procedure and derive a sparsity oracle inequality in the deterministic design case. In Section 3 the design can be either deterministic or random. We propose a modification of the first aggregation procedure for which we can establish a sparsity oracle inequality in probability for the true excess risk. Section 4 is devoted to the RJMCMC algorithm used to effectively implement our estimators. In Section 5 we carry out a simulation study and compare the performances of our methods with the Lasso. Finally Section 6 contains all the proofs of our results.

Sparsity Oracle Inequality in Expectation

Throughout this section, we assume that the design is deterministic and the noise variables W 1 , . . . , W n are i.i.d. gaussian N (0, σ 2 ).

For any J ⊂ {1, ..., p} and K > 0 define (2.1)

Θ(J) = θ ∈ R p : ∀j / ∈ J, θ j = 0 , and
(2.2) Θ K (J) = θ ∈ R p : |θ| 1 ≤ K and ∀j / ∈ J, θ j = 0 .
For the sake of simplicity we will write Θ K = Θ K ({1, ..., p}).

For any subset J ⊂ {1, ..., p} define θJ ∈ arg min where λ > 0 is the temperature parameter, π is the prior probability distribution on P({1, . . . , p}), the set of all subsets of {1, ..., p}, that is, for any J ∈ {1, ..., p}, π J ≥ 0 and J∈P({1,...,p}) π J = 1. In Section 4 we show that the estimator θn can be computed in reasonable time even when p is large using a MCMC scheme, namely, Hastings-Metropolis algorithm. The parameters π and λ must be tuned in a suitable way. The choice of π is discussed below. The choice of the temperature parameter λ is discussed in Section 5.

θ∈Θ(J) r(θ), where r(θ) = 1 n n i=1 Y i -f θ (X i ) 2 = Y -f θ 2 n with Y = (Y 1 , ..., Y n ) T .
We now state the main results of this section.

Proposition 1. Assume that the noise variables W 1 , . . . , W n are i.i.d. N (0, σ 2 ).

Then the aggregate θn defined by (2.3) with 0 < λ

n 4σ 2 satisfies (2.4) E r( θn ) min J∈Pn({1,...,p}) E[r( θJ )] + 1 λ log 1 π J .
Proposition 1 holds true for any prior π and is due to Leung and Barron [START_REF] Leung | Information theory and mixing least-squares regressions[END_REF]. In what follows, we exploit this result in order to establish a sharp sparsity oracle inequality for the aggregation procedure (2.3). We suggest the following prior. Fix α ∈ (0, 1). Define π as follows (2.5)

π J = α |J| n j=0 α j p |J| -1
, ∀J ∈ P({1, ..., p}).

We have the following theorem Theorem 1. Assume that the noise variables W 1 , . . . , W n are i.i.d. N (0, σ 2 ). Then the aggregate fn = f θn , with λ = n 4σ 2 and π taken as in (2.5), satisfies

(2.6) E fn -f 2 n min θ∈R p    f θ -f 2 n + σ 2 |J(θ)| n 4 log pe |J(θ)|α + 1 + 4σ 2 log 1 1-α n    ,
where for any θ ∈ R p J(θ) = {j : θ j = 0}.

Tsybakov [START_REF] Tsybakov | Optimal rates of aggregation[END_REF] introduced the notion of optimal rate of aggregation adapting existing tools from the minimax theory. In particular, the rate derived in Theorem 1 is the optimal rate of sparse linear aggregation and does not depend on the magnitude of the nonzero components of θ. This result can be compared with the sparsity oracle inequalities established in [START_REF] Bickel | Simultaneous analysis of lasso and dantzig selector[END_REF][START_REF] Candes | The dantzig selector: statistical estimation when p is much larger than n[END_REF][START_REF] Dalalyan | Aggregation by exponential weighting, sharp oracle inequalities and sparsity[END_REF] where the rates becomes very large if the nonzero components of θ take large values.

Sparsity Oracle Inequality in Probability

From now on, the design can be either deterministic or random. We make the following mild assumption:

L = max 1≤j≤M φ j ∞ < ∞.
We assume in this section that the noise variables are subgaussian. More precisely we have the following condition.

Assumption 1. The noise variables W 1 , . . . , W n are independent and independent of X 1 , . . . , X n . We assume also that there exist two known constants σ > 0 and ξ > 0 such that

E(W 2 i ) ≤ σ 2 ∀k ≥ 3, E(|W i | k ) ≤ σ 2 k!ξ k-2 .
The estimation method is a version of the Gibbs estimator introduced by Catoni [START_REF] Catoni | Statistical Learning Theory and Stochastic Optimization[END_REF][START_REF] Catoni | PAC-Bayesian Supervised Classification (The Thermodynamics of Statistical Learning)[END_REF]. Fix K ≥ 1 and c > 0. First we define the prior probability distribution as follows. For any J ⊂ {1, ..., p} let u J denote the uniform measure on Θ K+c (J). We define m(dθ) = J⊂{1,...,p}

π J u J (dθ)
with π taken as in (2.5).

We are now ready to define our estimator. For any λ > 0 we consider the probability measure ρλ admitting the following density w.r.t. the probability measure m

(3.1) dρ λ dm (θ) = e -λr(θ)
Θ K e -λr dm .

The aggregate fn is defined as follows

(3.2) fn = f θn , θn = θn (λ, m) = Θ K θ ρλ (dθ).
The practical computation of θn is discussed in Section 4. Define

C 1 = 8σ 2 + (2 f ∞ + L(2K + c)) 2 ∨ [8[ξ + (2 f ∞ + L(2K + c))]L(2K + c)] .
We can now state the main result of this section.

Theorem 2. Let Assumption 1 be satisfied. Take K > 1, c = n -1 and λ = λ * = n 2C1 . Assume that arg min θ∈R p R(θ) ∩ Θ K = ∅. Then we have, for any ε ∈ (0, 1) and any θ ∈ arg min θ∈R p R(θ) ∩ Θ K , with probability at least 1 -,

R( θn ) ≤ R( θ) + 3L 2 n 2 + 8C 1 n |J( θ)| log (K + c) + |J( θ)| log enp α|J( θ)| + log 2 ε(1 -α) .
This theorem improves upon previous results on the following points:

• Our result holds under mild conditions on the dictionary while majority of the results under the sparsity scenario impose stringent conditions on the dictionary (see, e.g, [START_REF] Bach | Model-consistent sparse estimation through the bootstrap[END_REF][START_REF] Bickel | Simultaneous analysis of lasso and dantzig selector[END_REF][START_REF] Candes | The dantzig selector: statistical estimation when p is much larger than n[END_REF]). • we state a sparsity oracle inequality in probability for the risk R(•) whereas previous results concern either the empirical risk or are given in expectation [START_REF] Dalalyan | Aggregation by exponential weighting, sharp oracle inequalities and sparsity[END_REF][START_REF] Juditsky | Learning by mirror averaging[END_REF][START_REF] Lounici | Generalized mirror averaging and d-convex aggregation[END_REF]. • Unlike mirror averaging or progressive mixture rules, satisfying similar inequalities in expectation, our estimator does not involve an averaging step. As a consequence, its computational complexity is significantly reduced as compared to those procedures with averaging step.

The choice λ = λ * comes from the optimization of a (rather pessimistic) upper bound on the risk R (see Inequality (6.7) in the proof of this theorem, page 18). However this choice is not necessarily the best choice in practice even though it gives the good order of magnitude for λ. Section 5 illustrates this point. The practitioner may use cross-validation to properly tune the temperature parameter.

Practical computation of the estimator

Practical computation of θn and θn , for a given temperature λ > 0, is delicate. Indeed, exact computation of these estimators requires considering all subsets Θ K (J), for any J ⊂ {1, ..., p}. Since there are 2 p such subsets, exact computation of θn or θn is not feasible for large p. However, since our estimators are defined as expectations of posterior distributions, we can approximate them via Monte Carlo computation. There exists an extensive literature on Monte Carlo computational methods, especially in Bayesian statistics where estimators can sometimes be expressed as expectations of posterior distribution, see e.g. Marin and Robert [START_REF] Marin | Bayesian Core: A practical approach to computational Bayesian analysis[END_REF]. A standard Markov Chains Monte Carlo (MCMC) algorithm such as Hastings-Metropolis can be used to compute θn since the prior π used to define this estimator is a discrete probability distribution. The computation of θn is more delicate. Indeed ρλ is absolutely continuous w.r.t. the measure m(dθ) which involves a mixture of Lebesgue measures on spaces of different dimensions. A way to proceed with such measures was proposed by Green [START_REF] Green | Reversible jump markov chain monte carlo computation and bayesian model determination[END_REF] under the name "Reversible Jump Markov Chain Monte Carlo", RJMCMC, and applied successfully in various problems of model selection like multiple change-point problems, image segmentation and partition models in [START_REF] Green | Reversible jump markov chain monte carlo computation and bayesian model determination[END_REF] or selection of the number of components in a mixture model in Green and Richardson [START_REF] Green | On bayesian analysis of mixtures with an unknown number of components[END_REF]. We propose to adapt this procedure to our setting to compute θn . w J = 1.

Hastings-Metropolis algorithm starts from an arbitrary value (say J (0) = ∅), a simple transition kernel k(•, •) on the set {J ⊂ {1, ..., p}, |J| ≤ n} and updates J (t) to J (t+1) using the following scheme:

• draw I (t) from k(J (t) , •); • take
J (t+1) =    I (t)
with probability α(J (t) , I (t) ) = min 1,

w I (t) k(I (t) ,J (t) ) w J (t) k(J (t) ,I (t) )
J (t) with probability 1 -α(J (t) , I (t) ).

We stop after T steps and compute our estimator as the mean of the θJ (t) :

θw,T,bo = 1 T -bo + 1 T t=bo θJ (t)
where, as usual in MCMC methods, we remove the bo first simulations (burn-in period).

This algorithm ensures that (J (t) ) t is a Markov chain with invariant probability distribution (w J ) J (see Marin and Robert [START_REF] Marin | Bayesian Core: A practical approach to computational Bayesian analysis[END_REF]). Here, the set {J ⊂ {1, ..., p}, |J| ≤ n} being finite, we just have to remark that the chain is irreducible and aperiodic to obtain the convergence of θw,T,bo to θ(w) (in probability).

In practice, we use the following kernel k:

k(J, •) = k + (J, •)1 {|J|=0} + k + (J, •) + k -(J, •) 2 1 {0<|J|<n} + k -(J, •)1 {|J|=n}
where k + (•, •) and k -(•, •) are two kernels that we define now. The kernel k + adds an element to J whereas k -removes one element from J. When we try to add an element, it is reasonable to consider first features that are the most correlated with the current residual. Similarly when we try to remove one element, we give priority the feature with the smallest coefficients in absolute value. Formally, we choose some parameter ζ > 0. We put, for j / ∈ J:

k + (J, J ∪ {j}) = e ζ|cj | h / ∈J e ζ|c h |
where c j is the coefficient of linear correlation between Y i -f θJ (X i )

1≤i≤n and (φ j (X i )) 1≤i≤n . And, for j ∈ J:

k -(J, J \ {j}) = e -ζ|( θJ )j | h∈J e -ζ|( θJ ) h | .
Remark 1. The rationale behind our choice for k + (•, •) is the following. If ζ = 0, then the above algorithm adds a new coordinate to the model uniformly at random among the set of unused coordinates. This procedure, although reasonable in theory, is not efficient in practice when p becomes large. Indeed for large p (say p = 10 4 and p 0 = 2) a large number of steps can be necessary before we select an interesting coordinate. On the other hand, for large ζ the above procedure selects at each step the coordinate to incorporate to the model in a greedy way, that is the coordinate corresponding to the most correlated element of the dictionary with the current residual. For more details on greedy algorithms, see Barron et al [START_REF] Barron | Adaptative approximation and learning by greedy algorithms[END_REF][START_REF] Huang | Risk of penalized least squares, greedy selection and l1 penalization for flexible function libraries[END_REF]. This procedure is computationally efficient and performs better in high-dimension than the former one with ζ = 0. However it can sometimes be trapped in a local minimum. Therefore using an intermediate value for ζ seems to give a good balance in practice between these two extreme cases.

4.2. RJMCMC algorithm and computation of θn . The RJMCMC algorithm proposed by Green [START_REF] Green | Reversible jump markov chain monte carlo computation and bayesian model determination[END_REF] is an application of the Hastings-Metropolis to the case of a measure absolutely continuous with relation to a more sophisticated distribution (in our case m). We use here this method to compute θn = Θ K θ ρλ (dθ). We start from θ (0) = 0 and then, at each step, update θ (t) to θ (t+1) using the transition kernel. Note that we need to define a kernel k admitting a density w.r.t. the measure m.

For the sake of simplicity, we denote by ρλ (•) the measure ρλ as well as its density with respect to m, this is a standard convention in the MCMC literature. We define now the RJMCMC algorithm:

• draw τ (t) from k(θ (t) , •); • take ϑ (t) = τ (t) with proba. α(θ (t) , τ (t) ) = min 1, ρλ (τ (t) )k(τ (t) ,θ (t) ) ρλ (τ (t) )k(θ (t) ,τ (t) ) θ (t) with proba. 1 -α(θ (t) , τ (t) )
;

• draw θ (t+1) from the distribution ρλ (dθ|θ ∈ Θ(J(ϑ (t+1) ))).

This algorithm ensures that (θ (t) ) t is a Markov chain with invariant probability distribution ρ λ , see [START_REF] Marin | Bayesian Core: A practical approach to computational Bayesian analysis[END_REF]. The last step ensures that we can make a move inside the current model even if the model change was rejected by the Hastings-Metropolis step.

We define now the kernel k:

k(θ, •) = k + (θ, •)1 {J(θ)=∅} + k + (θ, •) + k -(θ, •) 2 1 {0<|J(θ)|<n} + k -(θ, •)1 {|J(θ)|=n}}
where, for some ζ > 0,

k + (θ, dθ ) = j / ∈J(θ) e ζ|cj (θ)| h / ∈J(θ) e ζ|c h (θ)| ρλ (dθ |θ ∈ Θ(J(θ) ∪ {j}))
and

k -(θ, dθ ) = j∈J(θ) e -ζ|θj | h∈J(θ) e -ζ|θ h | ρλ (dθ |θ ∈ Θ(J(θ) \ {j})),
where c j (θ) is the coefficient of linear correlation between (Y i -f θ (X i )) 1≤i≤n and (φ j (X i )) 1≤i≤n .

Simulations

We compare in this section the exponential weights estimators θn and θn to the LASSO [START_REF] Tibshirani | Regression shrinkage and selection via the lasso[END_REF] and the Elastic Net [START_REF] Zou | Regularization and variable selection via the elastic net[END_REF] on a toy example. 5.1. Description of the experiments. We consider variants of the toy example in [START_REF] Tibshirani | Regression shrinkage and selection via the lasso[END_REF]:

∀i ∈ {1, . . . , n = 20}, Y i = θ, X i + ε i with X i ∈ X = R p , θ ∈ R p and the ε i are i.i.d. from a gaussian distribution with mean 0 and standard deviation σ.

The X i 's are i.i.d. (and independent from the ε i ) and drawn from the gaussian distribution with zero mean and variance matrix:

Σ(ρ) = ρ |i-j| i ∈ {1, . . . , p} j ∈ {1, . . . , p}
for some ρ ∈ [0, 1). Note that Tibshirani's toy example is set with p = 8 whereas we will consider here p ∈ {8, 30, 100, 1000}.

Define the regression vector as either θ = (3, 1.5, 0, 0, 2, 0, 0, 0, 0, 0, ...)

or θ = (5e -1 , 5e -2 , 5e -3 , 5e -4 , ...)
corresponding respectively to a «sparse situation», |J( θ)| = 3 whatever is the value of p, and to an approximately sparse situation, with |J( θ)| = p but θ well approximated in the l 1 norm by a vector θ ∈ R p with |J(θ )| p. We will take σ 2 respectively equal to 1 («low noise situation») and 3 («noisy case»); the value of ρ is fixed to 0. For the LASSO, the 'optimal' theoretical choice of µ is proportional to σ log(p)/n (see [START_REF] Bickel | Simultaneous analysis of lasso and dantzig selector[END_REF] for example). We see from Theorem 1 and 2 that the 'optimal' theoretical choice of λ for the exponential procedures is of the order n/σ 2 . Thus we take for our experiment Λ = (n/σ 2 ) × G, Λ = σ 2 log(p)/n × G, where G is defined as follows:

G = 0.01 × {(1.5) i , i = 1, ..., 15} ∪ {0}.
We compute for our exponential weights estimators the oracle quadratic risk

inf λ∈Λ |X( θn (λ, π) -θ)| 2 2 and inf λ∈Λ |X( θn (λ, m) -θ)| 2 2
, and similarly the oracle quadratic risks for the Lasso and the Elastic Net

inf µ∈Λ |X( θL n (µ) -θ)| 2 2 and inf µ,γ∈Λ |X( θEN n (µ, γ) -θ)| 2 2 .

Numerical results.

We perform every experiment 20 times and give the results for the sparse situation in Table 1 and for the approximately sparse situation in Table 2. Convergence of the estimators can be checked on Figure 1.

We can see on these experiments that the exponential weights estimators outperforms the LASSO and the Elastic Net in the low noise case σ = 1. When σ grows, the performances of our estimators are still better, but the difference is less significative; moreover, θn seems to become less stable (in particular Table 1, p = 30 and σ 2 = 3).

This simulation study clearly shows the advantage to use the exponential weights estimators, in particular θn , especially in the situation of approximate sparsity. As we mentioned in the introduction, the main advantage of the LASSO and the Elastic Table 1. Numerical results for the estimation of θ = (3, 1.5, 0, 0, 2, 0, 0, 0, 0, 0, ...). For each possible combination of σ and p, we report the median, mean and the standard deviation values of respectively Net is the computation time. When p becomes larger (p > 1000), the MCMC takes much longer to converge and the computation time becomes prohibitive. The strength of 1 -penalized estimators is that they can be computed for large values of the dimension p 10 7 in a reasonable amount of time. On the other hand, these penalized methods are inferior to the exponential weights procedures in term of statistical properties when we consider the prediction problem.

inf λ∈Λ |X( θn (λ, π) -θ)| 2 2 , inf λ∈Λ |X( θn (λ, m) - θ)| 2 2 , inf µ∈Λ |X( θL n (µ) -θ)|

Some comments on computation time. We can roughly analyze the computational complexity of our MCMC algorithm θn as follows:

Table 2. Results for θ = 5 × (e -1 , e -2 , ..., e -p ). For each possible combination of σ and p, we report the median, mean and the standard deviation values of respectively

inf λ∈Λ |X( θn (λ, π) - θ)| 2 2 , inf λ∈Λ |X( θn (λ, m) -θ)| 2 2 , inf µ∈Λ |X( θL n (µ) -θ)| 2 2 and inf µ,γ∈Λ |X( θEN n (µ, γ) -θ)| 2 2 .
σ (2) At each step t ≤ T , we have to choose which new component we want to add (or remove) from the current model. There are at most p possible choices, and for each choice j we have to compute the correlation between the vectors (Y i -f θJ (t) ) 1≤i≤n and (φ j (X i )) 1≤i≤n , this takes O(np) operations.

(3) Finally, at each step t we have to compute θJ (t) , this takes at most O(|J| 3 ) operations.

Finally, the number of operations is All the simulations were performed with the R software [START_REF]R: A Language and Environment for Statistical Computing[END_REF]. The code are available on request by e-mail.

O T (np + E λ [|J| 3 ]) where E λ [|J|] is
6. Proofs 6.1. Proofs of Section 2. This proof uses an argument from Leung and Barron [START_REF] Leung | Information theory and mixing least-squares regressions[END_REF].

Proof of Proposition 1. The mapping Y → fn (Y ) = ( fn (X 1 , Y ), . . . , fn (X n , Y )) T is clearly continuously differentiable by composition of elementary differentiable functions. For any subset J ⊂ {1, . . . , p} define

A J = (φ j (X i )) 1≤i≤n,j∈J , Σ J = 1 n A T J A J , Φ J (•) = (φ j (•)) j∈J and g J = e -λ Y -f J 2 n + 2σ 2 |J| n where f J (x, Y ) = 1 n Y T A J Σ + J Φ J (x)
∂ i f J (x, Y ) = 1 n Φ J (X i )Σ + J Φ J (x) T , (∂ i f J (X 1 , Y ), . . . , ∂ i f J (X n , Y ))Y = f J (X i , Y ),
and

n l=1 f J (X l , Y )∂ i f J (X l , Y ) = f J (X i , Y ).
Thus we have

∂ i (g J ) = -λ∂ i Y -f J 2 n g J = - 2λ n (Y i -f J (X i , Y )) - n l=1 ∂ i f J (X l , Y )(Y l -f J (X l , Y )) g J = - 2λ n (Y i -f J (X i , Y ))g J , Recall that fn (•, Y )) = J∈Pn({1,...,p}) π J g J f J (•, Y )
J∈Pn({1,...,p}) π J g J .

We have

∂ i fn (X i , Y ) = J∈Pn({1,...,p}) π J (∂ i (g J )f J (X i , Y ) + g J ∂ i (f J (X i , Y ))) J∈Pn({1,...,p}) π J g J - J∈Pn({1,...,p}) π J g J f J (X i , Y ) J∈Pn({1,...,p}) π J ∂ i (g J ) ( J∈Pn({1,...,p}) π J g J ) 2 = - 2λ n Y i fn + 2λ n J∈Pn({1,...,p}) f J (X i , Y ) 2 π J g J J∈Pn({1,...,p}) π J g J + 1 n J∈Pn({1,...,p}) Φ J (X i )Σ + J Φ J (X i ) T π J g J J∈Pn({1,...,p}) π J g J + 2λ n Y i fn (X i , Y ) - 2λ n f 2 n (X i , Y ) = 2λ n J∈Pn({1,...,p}) (f J (X i , Y ) -fn (X i , Y )) 2 π J g J J∈Pn({1,...,p}) π J g J + 1 n J∈Pn({1,...,p}) Φ J (X i )Σ + J Φ J (X i ) T π J g J J∈Pn({1,...,p}) π J g J ≥ 0. (6.1)
Consider the following estimator of the risk (6.2)

rn (Y ) = fn (Y ) -Y 2 n + 2σ 2 n n i=1 ∂ i fn (X i , Y ) -σ 2 .
Using an argument based on Stein's identity as in [START_REF] Lehmann | Theory of Point Estimation[END_REF] we now prove that

E[r n (Y )] = E fn (Y ) -f 2 n . We have E fn (Y ) -f 2 n = E fn (Y ) -Y 2 n + 2 n n i=1 W i ( fn (X i , Y ) -f (X i )) -σ 2 = E fn (Y ) -Y 2 n + 2 n n i=1 W i fn (X i , Y ) -σ 2 . (6.3) For z = (z 1 , . . . , z n ) T ∈ R n write F W,i (z) = j =i F W,i (z j ), where F W denotes the c.d.f. of the random variable W 1 . Since E(W i ) = 0 we have E W i fn (X i , Y ) = E W i Wi 0 ∂ i fn (X i , Y 1 , . . . , Y i-1 , f (X i ) + z, Y i+1 , . . . , Y n )dz = R n-1 R y y 0 ∂ i fn (X i , f + z)dz i dF W (y) dF W,i (z). (6.4)
In view of (6.1) we can apply Fubini's Theorem to the right-hand-side of (6.4). We obtain under the assumption W ∼ N (0, σ 2 ) that

R + y 0 ∂ i fn (X i , f + z)dz i dF W (y) = R + ∞ zi ydF W (y)∂ i fn (X i , f + z)dz i = R + σ 2 ∂ i fn (X i , f + z)dF W (z i ),
A Similar equality holds for the integral over R -. Thus we obtain

E W i fn (X i , Y ) = σ 2 E ∂ i fn (X i , Y ) .
Combining (6.2), (6.3) and the above display gives

E [r n (Y )] = E fn (Y ) -f 2 n . Since fn (•, Y ) is the expectation of f J (•, Y ) w.r.t. the probability distribution ∝ g•π, we have fn (•, Y ) -Y 2 n = J∈Pn({1,...,p}) f J (•, Y ) -Y 2 n -f J (•, Y ) -fn (Y ) 2 n g J π J
J∈Pn({1,...,p}) g J π J

.

For the sake of simplicity set f J = f J (•, Y ) and fn = fn (•, Y ). Combining (6.2), the above display and λ

f J -Y 2 n + n i=1 4λσ 2 n -1 f J -fn 2 n g J π J J∈Pn({1,...,p}) π J g J + 2σ 2 n 2 n i=1 J∈Pn({1,...,p}) Φ J (X i )Σ + J Φ J (X i ) T π J g J J∈Pn({1,...,p}) π J g J -σ 2
J∈Pn({1,...,p})

f J -Y 2 n + 2σ 2 n |J| g J π J -σ 2 .
By definition of g J we have

f J -Y 2 n + 2σ 2 |J| n = - 1 λ log g J J∈Pn({1,...,p}) g J π J - 1 λ log   J∈Pn({1,...,p}) g J π J   .
Integrating the above inequality w.r.t. the probability distribution 1 C g • π (where C = J∈Pn({1,...,p}) g J π J is the normalization factor) and using the fact that J∈Pn({1,...,p})

1 C g J π J log 1 C g J = K g • π C , π 0
as well as a convex duality argument (cf., e.g., [START_REF] Dembo | Large Deviation Techniques and Applications[END_REF], p. 264) we get rn (Y )

J∈Pn({1,...,p}) Y -f J 2 n + 2σ 2 n |J| π J + 1 λ K(π , π) -σ 2 ,
for all probability measure π on P({1, . . . , p}). Taking the expectation in the last inequality we get for any π

E fn -f 2 n = E[r n (Y )] J∈Pn({1,...,p}) E[ f J -Y 2 n ] + 2σ 2 n |J| π J + 1 λ K(π , π) -σ 2 J∈Pn({1,...,p}) E[ f J -f 2 n ] + 2 n n i=1 E[W i f J (X i , Y )] + 2σ 2 n |J| π J + 1 λ K(π , π) J∈Pn({1,...,p}) E[ f J -f 2 n ] + 4σ 2 n |J| π J + 1 λ K(π , π),
where we have used Stein's argument

E[W i f J (X i , Y )] = σ 2 E [∂ i f J (X i , Y )] and the fact that n i=1 ∂ i f J (X i , Y ) = 1
in the last line. Finally taking π in the set of Dirac distributions on the subset J of {1, . . . , p} yields the theorem.

Proof of Theorem 1. First note that any minimizer θ ∈ R p of the right-hand-side in (2.6) is such that |J(θ)| rank(A) n where we recall that A = (φ j (X i )) 1 i n,1 j p . Indeed, for any θ ∈ R p such that |J(θ)| > rank(A) we can construct a vector θ ∈ R p such that f θ = f θ and |J(θ )| rank(A) and the mapping x → x log epα x is nondecreasing on (0, p].

Next for any J ∈ P n ({1, . . . , p}) we have

E[ f J -f 2 n ] = min θ∈Θ(J) f θ -f 2 n + σ 2 |J| n = min θ∈Θ(J) f θ -f 2 n + σ 2 |J(θ)| n .

Thus min

J∈Pn({1,...,p})

E[ f J -f 2 n ] + 1 λ log 1 π J + σ 2 J n = min J∈Pn({1,...,p}) min θ∈Θ(J) f θ -f 2 n + 1 λ log 1 π J(θ) + σ 2 |J(θ)| n = min θ∈R p f θ -f 2 n + 1 λ log 1 π J(θ) + σ 2 |J(θ)| n .
Combining the above display with Proposition 1 and our definition of the prior π gives the result.

6.2. Proof of Theorem 2. We state below a version of Bernstein's inequality useful in the proof of Theorem 2. See Proposition 2.9 page 24 in [START_REF] Massart | Concentration Inequalities and Model Selection (Saint-Flour Summer School on Probability Theory[END_REF], more precisely Inequality (2.21).

Lemma 1. Let T 1 , ..., T n be independent real valued random variables. Let us assume that there is two constants v and w such that

n i=1 E[T 2 i ] ≤ v
and for all integers k ≥ 3,

n i=1 E (T i ) k + ≤ v k!w k-2 2 .
Then, for any ζ ∈ (0, 1/w),

E exp ζ n i=1 [T i -E(T i )] ≤ exp vζ 2 2(1 -wζ) .
Proof of Theorem 2. For any θ ∈ Θ K+c define the random variables

T i = T i (θ) = -(Y i -f θ (X i )) 2 + (Y i -fθ(X i )) 2 .
Note that these variables are independent. We have

n i=1 E[T 2 i ] = n i=1 E [2Y i -fθ(X i ) -f θ (X i )] 2 [fθ(X i ) -f θ (X i )] 2 = n i=1 E [2W i + 2f (X i ) -fθ(X i ) -f θ (X i )] 2 [fθ(X i ) -f θ (X i )] 2 ≤ n i=1 E 8W 2 i + 2(2 f ∞ + L(2K + c)) 2 [fθ(X i ) -f θ (X i )] 2 = n i=1 E 8W 2 i + 2(2 f ∞ + L(2K + c)) 2 E [fθ(X i ) -f θ (X i )] 2 ≤ n 8σ 2 + 2(2 f ∞ + L(2K + c)) 2 R(θ) -R( θ) =: v(θ, θ) = v,
where we have used in the last line that .

f θ -fθ 2 = R(θ) -R( θ). Next we have, for any integer k ≥ 3, that n i=1 E (T i ) k + ≤ n i=1 E |2Y i -fθ(X i ) -f θ (X i )| k |fθ(X i ) -f θ (X i )| k ≤ n i=1 E 2 2k-1 |W i | k + ( f ∞ + L(K + c/2) k |fθ(X i ) -f θ (X i )| k ≤ n i=1 E 2 2k-1 |W i | k + ( f ∞ + L(K + c/2) k [L(2K + c)] k-2 [fθ(X i ) -f θ (X i )] 2 ≤ 2 2k-1 σ 2 k!ξ k-2 + ( f ∞ + L(K + c/2)) k [L(2K+c)] k-2 n i=1 E [fθ(X i ) -f θ (X i )] ≤ (σ 2 k!ξ k-2 + ( f ∞ + L(K + c/2)) k )(4L(2K + c)) k-2 4(σ 2 + ( f ∞ + L(K + c/2)) 2 ) v ≤ 1 4 k!ξ k-2 + [ f ∞ + L(K + c/2)] k-2 [4L(2K + c)] k-2 v ≤ 2 4 k! (ξ + [ f ∞ + L(K + c/2)]) k-2 [4L(2K + c)] k-2v ≤ v k!w k-2 2
Set C = 8 σ 2 + [ f ∞ + L(K + c/2)] 2 .
For any ε > 0 the last display yields

E exp λ - λ 2 C 2n(1 -wλ n ) R(θ) -R( θ) + λ -r(θ) + r( θ) -log 2 ε ≤ ε 2 .
Integrating w.r.t. the probability distribution m(•) we get

E exp λ - λ 2 C 2n(1 -wλ n ) R(θ) -R( θ) + λ -r(θ) + r( θ) -log 2 ε m(dθ) ≤ ε 2 .
Next, Fubini's theorem gives Jensen's inequality yields

E exp λ -
E exp λ - λ 2 C 2n(1 -wλ n ) Rdρ λ -R( θ) + λ -rdρ λ + r( θ) -K(ρ λ , m) -log 2 ε ≤ ε 2 .
Now, using the basic inequality exp(x) ≥ 1 R+ (x) we get

P λ - λ 2 C 2n(1 -wλ n ) Rdρ λ -R( θ) + λ -rdρ λ + r( θ) -K(ρ λ , m) -log 2 ε ≥ 0 ≤ ε 2 .

Using Jensen's inequality again gives

Rdρ λ ≥ R θ ρλ (dθ) = R( θλ ).

Combining the last two displays we obtain P R( θλ ) -R( θ) ≤ rdρ λ -r( θ) + 1 λ K(ρ λ , m) + log 2 ε 1 -λC 2(n-wλ)

≥ 1 -ε 2 .

Now, using Lemma 1.1.3 in Catoni [START_REF] Catoni | PAC-Bayesian Supervised Classification (The Thermodynamics of Statistical Learning)[END_REF] we obtain that (6.5) P R( θλ ) -R( θ) ≤ inf

ρ∈M 1 + (Θ K+c ) rdρ -r( θ) + 1 λ K(ρ, m) + log 2 ε 1 -λC 2(n-wλ) ≥ 1 - ε 2 .
We now want to bound from above r(θ) -r( θ) by R(θ) -R( θ). Applying Lemma 1 to Ti (θ) = -T i (θ) and similar computations as above yield successively where we have used that 1 -λC 2(n-wλ) ≥ 1/2 and 1 + λC 2(n-wλ) ≤ 3/2.

E exp λ R( θ) -R(θ) + r(θ) -r( θ) ≤ exp

4. 1 .

 1 Computation of θn via Hastings-Metropolis sampling. In this subsection, we write a particular form of Hastings-Metropolis algorithm that will allow to compute any estimator of the form θ(w) = J∈Pn({1,...,p}) w J θJ where we already defined θJ ∈ arg min θ∈Θ(J) r(θ), and w J ≥ 0, ∀J ⊂ {1, ..., p} and J∈P({1,...,p})

5 .

 5 We fix ζ = 2 in the RJMCMC algorithm, α = 1/10, T = 12000 and bo = 2000. The LASSO and the Elastic Net are defined respectively by θL n = θL n (µ) = arg min θ∈R p [r(θ) + µ|θ| 1 ] , µ > 0 and θEN n = θEN n (µ, γ) = arg min θ∈R p r(θ) + µ|θ| 1 + γ|θ| 2 2 , µ, λ > 0.

Figure 1 .

 1 Figure 1. Convergence of the first coefficient of θn in an experiment with σ = 1, p = 8. We represent the first coordinate of 1 N -bo+1 N t=bo θI (t) as a function of N = b 0 , ..., T .

.

  the expectation of |J| under the aggregation distribution with temperature parameter λ E λ [|J|] = J∈Pn({1,...,p}) π J e -λ r( θJ )+ 2σ 2 |J| n |J| J∈Pn({1,...,p}) π J e -λ r( θJ )+ 2σ 2 |J| n For properly tuned λ, we observe E[|J|] |J(θ)|. We understand why the sparsity (or approximate sparsity) of the parameter has an important influence on the computation time. Consider for example the case p = 100 and n = 50. If |J(θ)| = 10 then n × p = 5000 > 1000 = |J(θ)| 3 whereas if |J(θ)| = 25 then n × p = 5000 < 15625 = |J(θ)| 3 .

n 4σ 2

 2 yields rn (Y ) = J∈Pn({1,...,p})

,

  with w := 8(ξ + [ f ∞ + L(K + c/2)])L(K + c/2).Next, for any λ ∈ (0, n/w) and θ ∈ Θ K+c , applying Lemma 1 with ζ = λ/n givesE exp λ R(θ) -R( θ) -r(θ) + r( θ) ≤ exp vλ 2 2n 2 (1 -wλ n )

λ 2 C 2 .E

 22 2n(1 -wλ n ) R(θ) -R( θ) + λ -r(θ) + r( θ) -log 2 ε m(dθ) ≤ ε exp λ -λ 2 C 2n(1 -wλ n ) R(θ) -R( θ) + λ -r(θ) + r( θ) -log dρ λ dm (θ) -log 2 ε ρλ (dθ) ≤ ε 2 .

vλ 2 2n 2 ε 2 .M 1 +≥ 1

 2211 (1 -wλ n ), and so for any (data-dependent) ρ,E exp λ + λ 2 C 2(n -wλ) -Rdρ + R( θ) + λ rdρ -r( θ)Combining (6.6) and (6.5) with a union bound argument givesP R( θλ ) -R( θ) (Θ K+c )is the set of all probability measures over Θ K+c . Now for any δ ∈ (0, c] taking ρ as the uniform probability measure on the set{t ∈ Θ(J( θ)) : |t -θ| 1 ≤ δ} ⊂ Θ K+c (J( θ)) gives P R( θλ ) ≤ R( -ε.Taking δ = c = n -1 and the inequality log p |J( θ)| ≤ |J( θ)| log pe |J( θ)| gives (6.7) P R( θλ ) ≤ R( )| log (K + c) + |J( θ)| log epn α|J( θ)| + log 2 ε(1 -α) ≥ 1 -ε Taking now λ = n/(2C 1 ) (where we recall that C 1 = C ∨ w) in (6.7) gives P R( θλ ) ≤ R( )| log (K + c) + |J( θ)| log enp α|J( θ)| + log 2 ε(1 -α) ≥ 1 -ε,

  Denote by P n ({1, . . . , p}) the set of all subsets of {1, . . . , p} containing at most n elements.

	The aggregate fn is defined as follows
	(2.3)	fn = f θn , θn = θn (λ, π) =	J∈Pn({1,...,p}) π J e J∈Pn({1,...,p}) π J e -λ r( θJ )+ -λ r( θJ )+ 2σ 2 |J| n 2σ 2 |J| n	θJ

  T , and Σ + J denotes the pseudo-inverse of Σ J . Denote by ∂ i the derivative w.r.t. Y i . Simple computations give


Christine Hamelin

Christine Salomon

Diane Cyr

Alice Gueguen

France Lert

Childhood sexual abuse and adult sexual health among indigenous Kanak

Introduction

Violence against women is increasingly recognized as a major public health concern in Oceania. Studies relating to this issue, mostly carried out in countries of the region with populations of predominantly European origin, have revealed the impact of abuse suffered by girls and women on their sexual and reproductive health [START_REF] Fanslow | Prevalence of child sexual abuse reported by a cross-sectional sample of New Zealand women[END_REF][START_REF] Fergusson | Childhood sexual abuse, adolescent sexual behaviors and sexual revictimization[END_REF]Secretariat of the Pacific Community, 2003;[START_REF] Taft | Violence against young Australian women and association with reproductive events: a cross sectional analysis of a national population sample[END_REF]). However, very few studies have described the long-term consequences of gender-based violence on indigenous women's sexual health, even though women within these communities have often been described as being at higher risk from abuse and in poorer health compared to other groups [START_REF] Anderson | Indigenous health in Australia, New Zealand, and the Pacific[END_REF][START_REF] Bramley | Disparities in Indigenous Health: A Cross-Country Comparison Between New Zealand and the United States[END_REF][START_REF] Salomon | HIV protection and screening in women in New Caledonia[END_REF]. This paper aims to analyze the associations between childhood sexual abuse (CSA) and adult sexual health among indigenous Kanak and non-Kanak women in New Caledonia.

Epidemiological studies have revealed an association between CSA and a long-lasting alteration in the capacity of victims to manage their fertility and sexuality. Studies focusing on sexual risk behavior in adult women are less concordant than those focusing on sexuality and reproductive health in adolescence [START_REF] Littleton | Sexual and physical abuse history and adult sexual risk behaviors: relationships among women and potential mediators[END_REF]. Nevertheless, numerous other studies suggest that, alongwith factors linked to a context of both gender inequalities and globally unfavorable social and economic conditions, adult sexual and reproductive health also depends on adverse personal experiences in childhood, especially CSA. Indeed, CSA has been found to be associated with reduced condom use [START_REF] Maman | The intersections of HIV and violence: directions for future research and interventions[END_REF][START_REF] Molitor | History of forced sex in association with drug use and sexual HIV risk behaviors, infection with STDs, and diagnostic medical care: Results from the Young Women Survey[END_REF], less contraceptive use [START_REF] Heise | A global overview of gender-based violence[END_REF], unintended pregnancy [START_REF] Dietz | Unintended pregnancy among adult women exposed to abuse or household dysfunction during their childhood[END_REF][START_REF] Steel | The association between childhood and adolescent sexual abuse and proxies for sexual risk behavior: a random sample of the general population of Sweden[END_REF], abortion [START_REF] Russo | Violence in the Lives of Women Having Abortions: Implications for Practice and Public Policy[END_REF][START_REF] Wingood | Child sexual abuse, HIV sexual risk, and gender relations of African-American women[END_REF] and sexually transmitted infections (STIs) [START_REF] Hillis | Adverse childhood experiences and sexually transmitted diseases in men and women: a retrospective study[END_REF][START_REF] Petrak | The association between abuse in childhood and STD/HIV risk behaviours in female genitourinary (GU) clinic attendees[END_REF]. Abused women were also found to be at significantly higher risk of experiencing dissatisfaction in their sex lives and of presenting sexual difficulties in adulthood [START_REF] Loeb | Child sexual abuse: associations with the sexual functioning of adolescents and adults[END_REF][START_REF] Mullen | The long-term impact of the physical, emotional, and sexual abuse of children: a community study[END_REF]. Finally, early sexual abuse has been found to be a major risk factor for sexual assaults occurring in adult life and for abuse suffered from partners [START_REF] Fergusson | Childhood sexual abuse, adolescent sexual behaviors and sexual revictimization[END_REF][START_REF] Fleming | The long-term impact of childhood sexual abuse in Australian women[END_REF][START_REF] Steel | The association between childhood and adolescent sexual abuse and proxies for sexual risk behavior: a random sample of the general population of Sweden[END_REF].

Among indigenous women, although the association between early sexual violence and heavy alcohol use has been investigated [START_REF] Hamelin | Childhood sexual abuse and adult binge drinking among Kanak women in New Caledonia[END_REF][START_REF] Koss | Adverse Childhood Exposures and Alcohol Dependance Among Seven Native American Tribes[END_REF][START_REF] Kunitz | Alcohol dependence and domestic violence as sequelae of abuse and conduct disorder in childhood[END_REF][START_REF] Libby | Childhood Physical and Sexual Abuse and Subsequent Alcohol and Drug Use Disorders in Two American-Indian Tribes[END_REF], very few studies have focused on sexual health indicators linked to childhood sexual abuse. Work carried out by [START_REF] Young | Survivors of sexual abuse: clinical, lifestyle and reproductive consequences[END_REF] in Canada, made the distinction between aboriginal and non-aboriginal women.

In both groups, a history of sexual abuse was associated with an increase in the probability of STIs, sexual intercourse before the age of 12 years, and multiple sexual partners. While no association was observed with the conjugal or social situation of indigenous women, divorce or separation from a partner, as well as unemployment was associated with a previous history of sexual abuse in non-indigenous women. [START_REF] Yuan | Risk factors for physical assault and rape among six Native American tribes[END_REF] studied physical assault and rape in adults from six U.S. Native tribes. They found that, as in the general population, childhood victimization was a predictor, whereas other predictors (e.g., level of tribal affiliation) were group specific.

New Caledonia is still a French ruled country, where the indigenous Kanak population is neither an overwhelming majority, as it is the case in the neighboring Melanesian states, nor a small minority as in Australia and New Zealand. The Kanaks represent 44% of the total population of 230 000 inhabitants while Europeans (i.e. born in France or of European descent) account for 34%, Polynesians 12% (Wallisians and Futunians, and Tahitians), Asians 5 % and persons from another community 5% (Institut Territorial de la Statistique et des Etudes Economiques, 2001). In the 1980s, a nationalist rebellion for Kanak independence forced concessions from France and a process of re-balance of power in favor of the Kanak community (Matignon Accords, 1989& Nouméa Accord, 1998). The political and economic transformations that have taken place since then have led to increasing social heterogeneity within the Kanak community and have fostered significant changes for women: consolidation of a female urban population, increased level of girls' education, subsequent better access to employment and opportunity to earn a personal income, and implementation of gender parity in politics [START_REF] Berman | The law on gender parity in politics in France and New Caledonia: a window into the future or more of the same? Oxford University Comparative Law Forum[END_REF][START_REF] Hamelin | Les gens de Nouméa. Mutations et permanences en milieu urbain[END_REF][START_REF] Salaün | From "primitive mentality" to "clash of cultures": stereotypes and indigenous underachievement in New Caledonian schools[END_REF]. However, these major ongoing changes are not sufficient to counterbalance more than 150 years of colonial domination.

Significant socioeconomic inequalities persist today. The Kanak population, who lives mainly in rural areas and islands, still holds a socioeconomically disadvantaged position in New Caledonian society. The 2004 census indicated that in the Northern Province and in the Loyalty Islands, where Kanaks are the overwhelming majority, the proportion of women aged 14 and over with no diploma was respectively 56% and 63 %. In the Southern Province, which is home to a mainly non-Kanak population, this proportion was 32% (Institut de la statistique et des études économiques, 2007). The non-Kanak population resides mostly in the Noumea metropolitan area but is quite heterogeneous regarding socio-economic characteristics and ethnicity. It is composed of successive migration waves, from the initial French convicts and settlers, and from the descendants of the Oceanian and Asian labor trade in the 19 th and first part of the 20 th century to more recent immigration from Polynesia and France [START_REF] Merle | Experiences coloniales : La Nouvelle-Calédonie, 1853-1920[END_REF]. The Europeans, particularly wealthy French expatriates, have the highest position in terms of education, employment, and economic resources. The Polynesian and Asian inhabitants are well represented in certain segments of the local economy (as construction workers or small entrepreneurs for example), though their socio-economic position is lower than the Europeans'.

Although there is a certain amount of cultural mixing between the different communities several distinct cultural contexts and sets of social features exist. In particular, male domination remains a central element of gender relations among Kanaks. This is mostly evident in the division of daily activities, and the confining of women within the domestic sphere and obligatory maternity, as well as in obstacles faced by women wishing to escape an unhappy union, since divorce is almost impossible in Kanak rules and separating from a partner means leaving the children with him. These patterns, the value placed on reproduction and the license previously given to men to appropriate women's reproductive capacities through marriage and to impose discipline, as well as the scale of gender violence, all contribute to explain the narrow margins within which Kanak women can negotiate their sexuality [START_REF] Berman | Kanak women and the colonial process[END_REF][START_REF] Salomon | La personne et le genre au centre nord de la Grande Terre (Nouvelle-Calédonie)[END_REF][START_REF] Salomon | Obligatory Maternity and Diminished Reproductive Autonomy in A'jië Paicî Kanak Societies[END_REF].

Prior to the present study (2002), only data from police, criminal justice or women's organisations were available regarding CSA without any information on ethnicity. Prison records indicated that among male inmates, 15.9% were sentenced for violence against adults or minors and 39% for rape or other forms of sexual assault [START_REF] Guillonneau | Les détenus outre-mer Paris: Ministère de la justice[END_REF]. In 2002, in a total population of 200 000, 200 sexual assaults and rapes of minors were reported to law enforcement officials (Les Nouvelles Calédoniennes July 1, 2003). The same year, two thirds of the victims seeking help from "SOS Sexual Violence", an association helping victims of abuse, were minors (Les Nouvelles Calédoniennes July 1, 2003). However, because childhood sexual abuses are rarely denounced, such data from administrative or associative sources largely underestimated CSA rates.

To date, reproductive and sexual health has not been well-documented in New Caledonia and data on the ethnic community are even rarer. An existing system monitoring STIs, providing underestimated figures, nevertheless indicates an average annual incidence rate between 1997 and 2007 of 30 per 10 000 inhabitants, 103.4 per 10 000 in the 20 -24 years age group. The average incidence appeared to be five times higher in women compared to men (Direction des Affaires Sanitaires et Sociales, 2007). The scarce ethnic data confirm the existence of strong inequalities. Indeed, one survey conducted in 1999 indicated a significant difference between communities in the frequency of pregnancies without medical follow-up, with Kanak women showing the highest rate (25.7%) and European and Asian women showing the lowest (4.4%) [START_REF] Barny | Etude sur les grossesses peu ou pas suivies en Nouvelle-Calédonie[END_REF]. In addition, the number of Kanak women declaring their last pregnancy as unwanted is double that of European women (32% versus 16%) even though the rate of abortion does not differ significantly between communities [START_REF] Salomon | HIV protection and screening in women in New Caledonia[END_REF].

The present analysis examines the association between early experiences of sexual abuse and adult sexual health among both Kanak women and non-Kanak women. Such a comparative perpective allows examining whether or not CSA is associated with adult sexual health in New Caledonia irrespective of ethnic group membership and other sociodemographic characteristics. Thus, it affords the opportunity to explore whether poor sexual health indicators among adult Kanak women might be linked to exposure to sexual abuse during childhood.

Methods

Participants and Procedure

We used data obtained from a general population survey conducted between November 2002 and August 2003 in New Caledonia among women aged 18-54. The sample was drawn from the electoral rolls where most New Caledonian women are registered (88.2% in 2002 in this age group), due to the political issues surrounding a self-determination referendum expected in 2014. Sampling from electoral rolls was stratified by age to better reflect the age structure of the New-Caledonian population. The sampling method resulted in all subjects having an equal weight of 1. To be sure of obtaining around 1000 respondents, we drew an age-stratified sample of 4000 women at random. These 4000 subjects served as a pool from which women were drawn in successive waves to be contacted, until the required number of interviewees was reached. In all, 1679 women were drawn from the pool of 4000 for interviewers to contact. A letter was sent to these 1679 women to announce that an interviewer would soon contact them either by phone or directly at home. The letter clearly stated their right to decline the interview, to subsequently change their mind before or during the interview and to access and rectify the data. This letter presented the survey as a study on the living conditions and security of New Caledonian women and described the questions as related to childhood and adolescence experiences and to current family life. Violence, including childhood sexual abuse, was not explicitly identified as one of the survey topics. A total of 580 women could not be located or had moved outside the area of residence registered on the electoral rolls. Of the 1099 successfully contacted, 1012 (92.1%) agreed to participate.

Of the women who did not participate, 5 % refused to take part and 2.9 % missed their appointments. Less than 1% cut the interview short, mostly before the questions on violence were asked. Only the women of reproductive age (aged from 18 to 49 years) and sexually active during the preceding twelve months were included in the present analyses (Kanak women n = 329, non-Kanak women n = 436).

Data were collected anonymously using a standardized questionnaire in face-to-face interviews that lasted an average of one hour. The interviews were conducted by 15 female interviewers belonging to the various ethnic communities of New Caledonia (Kanak, European, Polynesian, and Asian) and trained by the two main investigators of the study.

Women were interviewed in complete privacy, for the main part in their homes or in another place chosen by them. The first part of the questionnaire concerned women's social and demographic situation. Ethnic classification was based on self-definition, according to the categories used in the 1996 New Caledonia Census (Melanesian, European, Wallisian or Futunian, Tahitian, Indonesian, Vietnamese, Other). In the present analysis the Kanak group consisted of women who identified themselves as Melanesian while the non-Kanak group aggregated women who identified themselves as European, Wallisian or Futunian, Tahitian, Indonesian, Vietnamese or from another community. The second part of the interview concerned women's health, including sexual and reproductive health. A third part enquired about any abuse suffered over the last 12 months leading up to the interview. Finally, the last part related to any physical abuse suffered from the age of 15 years onwards and sexual abuse suffered during childhood and later. All women who experienced sexual abuse were asked to identify the perpetrator of the single event or of the first event if repeated abuse was reported.

The study was approved by the French data protection authority (Commission Nationale Informatique et Libertés). It complied with the ethical guidelines for confidentiality formulated by the WHO [START_REF] Watts | Putting women first: Ethical and safety recommandations for research on domestic violence against women[END_REF].

Measures Childhood sexual abuse

Childhood sexual abuse was assessed using two questions about lifetime sexual abuse: "During your life, has anyone ever fondled you sexually when you did not want to, or unsuccessfully tried to have sexual intercourse with you?" , "During your life, has anyone ever forced you to have sexual intercourse against your will?". In both cases, women were asked their age at the time of the event or at the first and last time when repeated abuse was reported. Thus, the indicator of childhood sexual abuse was defined as having experienced sexual fondling, attempted rape or rape, one or more times before the age of 15. The age of 15 was used as cutpoint because of differences in the characteristics of abuse before age 15 and abuse at age 15 and older. In addition, the age of 15 was the upper limit often used in international studies to define sexual abuse experienced during childhood (Garcia-Moreno, Jansen, Ellsberg, [START_REF] Garcia-Moreno | WHO multi-country study on women's health and domestic violence against women: initial results on prevalence, health outcomes and women's responses[END_REF].

Sexual health indicators

Five indicators of sexual health were examined:

Regarding the use of contraception, women who answered yes to the question "Currently, do you use a method of contraception to avoid being pregnant?" were asked to specify the method. Women using no contraception or who had resorted to traditional methods (withdrawal, temperature, local plant remedies) were compared to those using modern contraception (contraceptive pill, intrauterine device, condom, implant, tubal ligation). A total of 216 (109 Kanak, and 107 non-Kanak) women were excluded from analyses examining the association between CSA and contraception because they indicated that they were infertile, pregnant, or wanted to become pregnant and thus were not in need of contraception.

Women who reported ever having been pregnant were questioned about whether and when they had an abortion. Women who declared an abortion during the last five years were compared to the others.

The variable STIs identified those women who answered yes to the question: "During the last 12 months, did you suffer from a sexually transmissible disease?"

Women were asked how often they had had undesired sexual intercourse in the last twelve months with their intimate partner. The exact wording of the survey question was: "During the last 12 months, did you accept sexual intercourse without really wanting it?" Answers were split into two categories: often versus never, rarely or sometimes.

Several questions were used to document adult sexual violence, depending on the perpetrator. Two questions assessed intimate parter/ex-partner's sexual violence: "During the last 12 months, did your partner/ex-partner force you to have sexual intercourse when you did not want to?" and "During the last 12 months, did your partner/ex-partner force you to do or to submit to sexual acts when you did not want to?" Regarding close relatives, acquaintances and strangers two questions were asked for each: "During the last 12 months, did a close relative (acquaintances/stranger) try unsuccessfully to have sexual intercourse with you, or to force you to engage in sexual acts against your will?" and "During the last 12 months, did a relative (acquaintance/stranger) force you to have sexual intercourse against your will?".

Adult sexual violence covered any of the situations described above from any perpetrator (intimate partner /ex-partner, close relative, acquaintance and stranger).

Statistical analysis

We studied the associations between CSA and indicators of adult sexual health, adjusting for age and marital status, both of which are variables directly associated with sexual behavior. Age was divided into three classes: 18-25 years, 26-37 years and 38-49 years. Three categories were used to define marital status: married or living together as a couple, being in a stable relationship (for at least 6 months) though not living together, and no stable relationship over the last year. Women's sexual health appears also influenced by socioeconomic and educational factors [START_REF] Wellings | Sexual behaviour in context: a global perspective[END_REF]. Educational level and personal income level were considered and included in logistic regression models. The highest level of educational achievement was divided into 3 categories: none or elementary school qualification, below baccalaureate (vocational qualifications at the end of junior high), Baccalaureate or higher, the Baccalaureate being the diploma obtained upon completing the high school curriculum. In New Caledonia, the social value of this qualification has historically been and remains important, and is an advantage for entering the paid workforce.

Personal income was divided into 2 categories taking as a cut point half the legal minimum wage: 335 Euros/month or less versus more than 335 Euros/month. The question on economic resources included revenue from employment, benefits and allowances awarded by the state, and money earned selling fish or cultivated products at the road side or at the market.

Occupational status was classified into 3 categories : employed, housewive or subsistence farmer, and unemployed, student or other.

We built separate models for Kanak and non-Kanak women. Descriptive analyses were performed to obtain information regarding the socio-demographic characteristics of both sub-samples. Bivariate Chi-square analyses were conducted to examine differences between Kanak and non-Kanak women in prevalence rates of CSA and adult sexual outcomes. No use of modern contraception, abortion, STIs, undesired sexual intercourse and adult sexual violence were taken as dependent variables. Separate logistic regression models assessed the association between CSA and each indicator of adult sexual health after adjusting for all covariates in Kanak and non-Kanak group. Occupational status was not included in multivariate models because of colinearity with personal income. All analyses were performed using Statistical Analysis Software (SAS v9.1, SAS Institute Inc, Cary, NC, USA).

Results

Socio-demographic characteristics

The average age of the Kanak women sub-sample was 32 years (SD 8.76); it was 33.5 years (SD 8.24) for the non-Kanak sub-sample. In both groups, most women were in a stable relationship but Kanak women were more likely not to live with their partner (Table 1). The groups differed substantially in terms of educational level and economic resources. Only 10.9% of Kanak women obtained the Baccalaureate compared with more than 40% of non-Kanak women. Over half of the Kanak women had no personal income or were earning a very small personal income. The proportion of non-Kanak women in the same situation was 27.3%. Regarding occupational status, 46.5% of Kanak women were employed (68.3 % among non-Kanak women), 39.8% declared being housewives or subsistance farmers (18.6% among non-Kanak women), 13.9% were unemployed, student or in another situation (12.6% among non-Kanak women).

Prevalence and characteristics of Childhood Sexual Abuse

A non significant difference between Kanak and non-Kanak women was found for CSA, the prevalence being respectively 11.8 and 14.4% (Table 2). Among women having suffered from sexual abuse, 38.5% of Kanaks and 22.2% of non-Kanaks had experienced rape. In both groups, most victims knew their aggressor. Nevertheless, the proportion of close relatives among the agressors appeared higher in Kanak (66.7%) than in non-Kanak women (42.8%). A non significant difference between Kanak and non-Kanak women was found for the frequency of abuse. b Women declared infertile and those pregnant or with a desire to be so at the time of interview were excluded from analyses on contraception (n =220 for Kanak women, and n = 329 for non-Kanak women)

Prevalence of sexual health indicators in adulthood

Failure to use modern contraception was more frequent among Kanak women than among non-Kanak women, while no significant difference between the two groups was observed for abortion (Table 2). Having been diagnosed for STIs in the last 12 months was more often reported by Kanak women compared to non-Kanak women (6.1% versus 1.2%).

Nearly one in three Kanak women reported regularly having undesired sexual intercourse with their intimate partner while this proportion was 17.7% among non-Kanaks. Finally, in the year leading up to the survey, 18.5% of Kanak women had suffered some form of sexual violence compared to 3.9% of non-Kanak women. In both groups, this was mainly reported as occurring within the couple.

Association between CSA and sexual health indicators in adulthood

Table 3 presents the results obtained from bivariate and multivariate analyses between these indicators and CSA in Kanak and non-Kanak women.

In Kanak women, failure to use modern contraception was less likely among women with a CSA history; Even though failure to use of modern contraception also correlated with age and personal income, its association with CSA did not change after adjustment for those two variables (Adjusted OR: 0.2, 95% CI: 0.07-0.8). While abortion was associated with a higher educational level, no correlation with CSA was observed. Kanak women who had suffered CSA were more likely to declare a recent STI. The association remained significant and very strong when adjusted (Adjusted OR: 3.8, 95% CI: 1.3-10.9). Regularly having undesired sexual intercourse with an intimate partner was found to be associated with CSA, age at the time of the survey and marital status. After adjusting for covariates, the association between CSA and non-desired sexual intercourse remained significant for this group and even increased (Adjusted OR: 3.4, 95% CI: 1.6-7.2). A strong relation was also found between CSA and experience of sexual violence in the 12 months leading up to the survey. The adjusted odds ratio for adult sexual violence was five times higher among Kanak women who had suffered CSA, compared to those who had not (Adjusted OR: 5.2, 95% CI: 2.4-11.1).

Among non-Kanak women, failure to use modern contraception was strongly correlated with low education, poverty and unstable intimate relationships but not with CSA.

A significant adjusted association between CSA and abortion was observed (Adjusted OR:

3.1, 95% CI: 1.1-8.8). Abortion was also associated with a low level of education. Regarding STIs in this group, the number of cases was too small to allow for bivariate and multivariate analysis (n = 5). While no significant association was observed between undesired sexual intercourse and CSA, women with a low level of education were more likely to have had undesired sexual intercourse with an intimate partner compared with those who had a higher educational level. Finally, in bivariate analysis, adult sexual violence was associated with age and income but not with CSA. After adjusting for covariates, the odds-ratio for CSA increased from 1.9 to 2.6 but did not reach statistical significance. The adjusted odds-ratio for age decreased and was no longer statistically significant. Only women with low personal income were still at significant risk for adult sexual violence.

Discussion

The results obtained in the present study suggest that relationships between CSA and adult sexual health problems differ between indigenous women and other women in New

Caledonia. With the exception of abortion, Kanak women's sexual and reproductive health is strongly associated with a history of childhood sexual abuse. Indeed, in this community, CSA considerably increases the risk of STIs, of non-desired sexual intercourse with an intimate partner and of experience of adult sexual violence, even after adjustment for potentially confounding variables (age, marital status, and educational and income level). But, whereas three outcomes -STIs, non-desired sexual intercourse and adult sexual violenceindicate an increased vulnerability associated with CSA, the fourth one -failure to use modern contraception displays a reverse association. In the non-Kanak group, only abortion appears significantly associated with CSA.

A number of limitations of this study have to be acknowledged. First, because of the low prevalence of adult sexual health problems among non-Kanak women, the sample size was not sufficient to detect an effect of CSA in this group. This was especially the case regarding the association between CSA and sexual violence experienced in adulthood ( 17cases reported in 436 women). Although the OR of adult sexual violence was also elevated in this group (Adjusted OR: 2.6, 95% CI: 0.7-9.0), the association was not statistically significant and the confidence interval was large. Secondly, the restricted size of the two subsamples did not allow us to take into account the characteristics of the early sexual abuse, which are known to influence long term health: age at onset of abuse, relationship to offender, duration and frequency of abuse, non-penetrative abuse or forced intercourse with penetration [START_REF] Beitchman | A review of the long-term effects of child sexual abuse[END_REF][START_REF] Fleming | The long-term impact of childhood sexual abuse in Australian women[END_REF][START_REF] Parillo | Association between early sexual abuse and adult HIV-risky sexual behaviors among community-recruited women[END_REF].

In addition, this retrospective study carried out among adult women revealed the associations, but not the causal pathway, between CSA and adult sexual health. Various factors not documented in this study such as psychiatric disorders [START_REF] Steel | The association between childhood and adolescent sexual abuse and proxies for sexual risk behavior: a random sample of the general population of Sweden[END_REF] or the characteristics of the partner [START_REF] Testa | Childhood sexual abuse, relationship satisfaction, and sexual risk taking in a community sample of women[END_REF], might be involved in the observed association between CSA and adult sexual health. The effects of potentially confounding factors could not be taken into account considering the limited data on family background and on other adverse childhood experiences, notably childhood physical and emotional abuse [START_REF] Mullen | The long-term impact of the physical, emotional, and sexual abuse of children: a community study[END_REF]. It has been shown that women exposed to CSA tend to have an earlier onset of consensual sexual intercourse with this, in turn, increasing risks of unprotected intercourse, multiple partnerships, and other forms of sexual risk taking later on [START_REF] Fergusson | Childhood sexual abuse, adolescent sexual behaviors and sexual revictimization[END_REF]. This association cannot be properly assessed in the present study: for some women the age at the first reported sexual experience corresponds to the age the CSA occurred. Thus, the first sexual intercourse reported may be the CSA itself.

Additionally, we have to consider the possibility that sexual activity at the time of data collection was associated with prior victimization history, which would introduce a bias in an analysis restricted to sexually active women. Actually, in both groups, sexual activity during the last 12 months was not associated with childhood sexual abuse (Age-adjusted OR: 0.9, 95% CI: 0.4-1.9 for Kanak women, and age-adjusted OR: 06, 95% CI: 0.3-1.3 for non-Kanak women).

To minimize under-reporting and to ease recollection and disclosure of early sexual abuse, the questions about CSA in our study were asked after those relating to the women's current living conditions and life history. Thus, trust gained during the interview might have facilitated answering such sensitive questions. But these techniques do not completely prevent under-reporting of CSA [START_REF] Watts | Violence against women: global scope and magnitude[END_REF][START_REF] Williams | Validity of women's self-reports of documented child sexual abuse[END_REF].

This under-reporting could lead to underestimating the observed associations.

This study also has a number of strengths. It is the first one to document prevalence of childhood sexual abuse in the general female population in New Caledonia and to provide sexual health data according to ethnic membership. In non-Kanak women, all sexual health problems were correlated to a low social status (i.e. low personal income or low educationnal level). Conversely, among Kanak women, STIs, undesired sexual intercourse with a partner and adult sexual violence were found to be unrelated to socio-economic factors. This lack of assocation between socio-economic characteristics and sexual health outcomes may reflect quite a narrow variability in education and income level within the Kanak community.

Because socioeconomic inequalities, ethnic divisions and disparities in health are tightly intertwined, Kanak women as a group are in the least favourable situation, in terms of social position, sexual heatlh and gender violence in adulthood. Thus, ethnicity still determines large differences regarding sexual health in New Caledonia today.

Despite this, the prevalence of sexual abuse during childhood was not found to significantly differ across the two groups in the study. This contrasts with the situation in neighboring Pacific countries, where higher frequencies were observed in indigenous minorities compared to the general population. The mean rate in Australia was 12%, but information on community was not available (Australian Bureau of Statistics, 2006).

However, the overexposure of Aboriginal girls to sexual abuse has often been stressed (P. [START_REF] Anderson | Report of the Northern Territory Board of Inquiry into the Protection of Aboriginal Children from Sexual Abuse Darwin: Nothern Territory Governement[END_REF][START_REF] Stanley | Child Abuse and Family Violence in Aboriginal Communities -Exploring Child Sexual Abuse in Wester Australia[END_REF]. In New Zealand, according to the definition used by the WHO, 23.5% of urban women and 28.2% of rural women were sexually abused before the age of 15 years. In both urban and rural areas, rates were higher among Maori women [START_REF] Fanslow | Prevalence of child sexual abuse reported by a cross-sectional sample of New Zealand women[END_REF].

In our study, the possibility of greater underreporting in Kanak women than in non-Kanak women has to be considered. As shown in a previous article, older and rural Kanak women were found much more inclined than their younger and urban counterparts to find a justification for male physical and sexual violence [START_REF] Salomon | Challenging Violence: Kanak Women Renegotiating Gender Relations in New Caledonia[END_REF] and rural Kanak women reported CSA significantly less often than their urban counterparts (10.4% versus 21.4%, p = 0.039). It is known that variations in reporting rates are more likely to reflect changes in social tolerance and agency practice than changes in the prevalence of violence [START_REF] Bajos | Les violences sexuelles en France : quand la parole se libère[END_REF][START_REF] Pereda | The prevalence of child sexual abuse in community and students samples: A meta-analysis[END_REF]. Such generational and residential differences were not recorded in the non-Kanak group in which the social process of rejection of gender violence, especially against young girls, seems to have been engaged for longer.

Even though the rate of CSA may not systematically vary between any two communities, the circumstances of this abuse, the level of tolerance or disapproval associated with it, and the effects of disclosure are susceptible to considerable variation, dependent on the socio-cultural context of each community, and thus influence the impact in adulthood in different ways [START_REF] Loeb | Child sexual abuse: associations with the sexual functioning of adolescents and adults[END_REF]. The results of the present study highlight the consequences of early female sexual abuse in the indigenous Kanak community where gender violence in adulthood and women's sexual vulnerability are particularly widespread. The overexposure to adult sexual violence, especially from an intimate partner, among female victims of CSA has been documented by numerous studies carried out in different regions, including Oceania. In Australia, the probability of experiencing domestic abuse and rape in adulthood was twice as high in women who had suffered CSA compared to those who had not [START_REF] Fleming | The long-term impact of childhood sexual abuse in Australian women[END_REF].

Similarly, in New Zealand, women reporting CSA were about twice as likely to have experienced sexual or physical abuse perpetrated by an intimate partner in the previous 12 months [START_REF] Fanslow | Prevalence of child sexual abuse reported by a cross-sectional sample of New Zealand women[END_REF]. Still, the association observed in Kanak women seems particularly strong, since those who suffered CSA were five times more likely to experience adult sexual violence than those who had not suffered CSA.

The strong association between CSA and STIs in Kanak women points to an increased likelihood of sexual risk behaviors in adult Kanak victims of CSA. This finding suggests that women who have been sexually abused in childhood encounter more difficulties negotiating the use of condoms by male partners in an effort to protect themselves [START_REF] Hillis | Adverse childhood experiences and sexually transmitted diseases in men and women: a retrospective study[END_REF][START_REF] Petrak | The association between abuse in childhood and STD/HIV risk behaviours in female genitourinary (GU) clinic attendees[END_REF]. In New Caledonia, where the margins within which Kanak women can negotiate their sexuality are narrow and where condom use is viewed as a "male business" especially in rural regions and among the less educated [START_REF] Bougerol | Représentations du sida et gestion du risque chez les femmes Kanakes de Nouvelle-Calédonie[END_REF], female child sexual abuse appears as an additional burden in regard to decision-making and management of sexuality later on.

The results also show that modern contraceptive use was less frequent among Kanak women than among non-Kanak women. The lower level of modern contraceptive use in Kanak women could be a consequence of a lack of autonomy in their reproductive lives due to strong gender inequalities. It might also be related to poor accessibility of medical facilities as shown by data on delayed onset of prenatal care among Kanak women [START_REF] Barny | Etude sur les grossesses peu ou pas suivies en Nouvelle-Calédonie[END_REF]. The positive association between CSA and the use of modern contraception in the Kanak group is thus all the more unexpected, especially as this association was not found in the non-Kanak group. Among Kanak women, modern contraceptive methods have only become widespread over the last 15 years and social norms require women to give birth to several children. To make sure that the observed association was not linked to a confounding effect (already having children), complementary analysis was carried adjusting for the number of children (0 versus 1 or more). The known relationship between the use of contraception and the number of children was also observed here (OR: 2.8, 95% CI: 1.2-6.7) but did not significantly modify the association between CSA and contraception use (Adjusted OR: 0.3, 95% CI: 0.07-0.9). The small size of the sub-samples did not allow us to conduct separate analyses for each contraception method. Of Kanak women using modern contraception, 50.6% were on the contraceptive pill, 28% had an intrauterine device, 13% used condoms, and 8.4% another method (implant or tubal ligation). Of non-Kanak women using modern contraception, 67.6%

were on the contraceptive pill, 22.9% had an intrauterine device, 3.8% used condoms and 5.7% another method. Thus, in both groups, contraception mostly captured the use of the pill or of an intrauterine device.

To our knowledge, little research investigates directly the association between sexual violence and contraception use. Some studies have shown reduced contraception use among women having suffered CSA compared to others [START_REF] Boyer | Sexual abuse as a factor in adolescent pregnancy and child maltreatment[END_REF][START_REF] Heise | A global overview of gender-based violence[END_REF].

Inconsistantly, among both indigenous and non-indigenous women in Canada, no association has been found between a history of sexual abuse and the use of oral contraception. However, in the non-indigenous group, the use of an intrauterine device was significantly more frequent among those who had suffered sexual abuse compared to those who had not. This association has not been found among indigenous women [START_REF] Young | Survivors of sexual abuse: clinical, lifestyle and reproductive consequences[END_REF].

The relationship between sexual violence and contraception appears to be particularly complex, all the more so in a society strongly valuing fertility. In New Caledonia, the synthetic fertility index is 2.2 children per woman (Direction des Affaires Sanitaires et Sociales, 2007), but this index does not reflect the situation in each community. Indeed data collected in this New Caledonian study showed large differences: Kanak women of 40 years of age or older had on average 4.8 children (compared to 3.8 for Polynesian women and 2.2

for European women and those from other communities).

The result on the relationship between CSA and contraception among Kanak women echoes a recent study on intimate partner violence carried out in New Zealand: although refusal of contraception by partners was significantly more important among women who had previously experienced domestic abuse, these women were also more likely than other women to have already used contraception [START_REF] Fanslow | Contraceptive use and associations with intimate partner violence among a population-based sample of New Zealand women[END_REF]. It could be that Kanak women victims of CSA, who are also more at risk of adult sexual violence, distance themselves from the reproductive social norm of their community and use medically proven forms of contraception which they can control more frequently than other Kanak women.

However, this assumption is not entirely consistent with the lack of association between CSA and abortion in the Kanak women. Several past studies have reported an association between abortion and a history of sexual abuse [START_REF] Russo | Violence in the Lives of Women Having Abortions: Implications for Practice and Public Policy[END_REF][START_REF] Wingood | Child sexual abuse, HIV sexual risk, and gender relations of African-American women[END_REF], as observed here in non-Kanak women. In the Kanak community, we also observed that a high level of education increased the likelihood of abortion. These findings suggest problems in the accessibility of abortion, with poor access among uneducated Kanak women, who represent a large majority of the Kanak group and who live far from the urban aera where all abortion clinics are located. Further analysis shows that among Kanak women who had an unintented pregnancy, 18% wishing an abortion was unable to obtain one, compared to 4% of non-Kanak women in the same situation.

Conclusion

This study shows that in all ethnic communities of New Caledonia, a history of child sexual abuse is not rare among women. The findings also shed light on the long term consequences of CSA in women. However, the comparison between indigenous Kanak women and other New Caledonian women suggests that the effect of CSA may differ according to ethnic membership and subsequent social stratification and gender norms. It seems all the more important to underline the extent and the impact of CSA since it was previously unrecognized: more than half (54%) of Kanak women and 38% of non-Kanak women who declared having suffered CSA before the age of 15 in the current survey had never disclosed their experience.

The results suggest a range of implications for clinical practice and prevention programs. Sexual risk-taking behavior and violence appear to be multidimensional public health concerns, calling for comprehensive programs that include awareness campaigns, education targeting both the male and female population, and care. In a country where distinct cultural contexts and sets of social features exist, appropriate prevention programs on violence and sexual health using diversified indicators to match the cultural and social heterogeneity are needed. Anthropological studies could help to further understand this issue. Interventions must address the complex factors contributing to a long-lasting alteration in the capacity of women to manage their sexuality. Since early sexual violence often remains a secret, a possible history of violence should be taken into account when assessing individual needs of women with sexual-health related problems. More generally, efforts to break the silence around violence against girls and establish a stronger foundation are required in New Caledonia.

Table 1

 1 Socio-demographic characteristics of Kanak and non-Kanak women

a  2 test

a Women declared infertile and those pregnant or with a desire to be so at the time of interview were excluded from analyses of the link between CSA and contraception (n =220 for Kanak women and n = 329 for Non Kanak women) Table 3 Childhood sexual abuse and sexual health outcomes in adulthood among Kanak (n = 329) and non-Kanak women (n = 436). Crude and adjusted odds ratios. 3) 26-37 years 0.5 (0.2-1.0) 0.4 (0.2-0.9) 1.8 (0.9-3.6) 1.9 (0.9-4.0) 2.6 (0.7-10.0) 2.2 (0.6-8.9) 3.8 (0.8-17.5) 4.0 (0.9-18.6) 38-49 years

Failure to use modern contraception

3) 2.5 (0.8-7.9) No stable partner during the year 2.2 (0.8-5.8) 2.0 (0.7-5.8) 3.9 (1.9-8.0) 5.4 (2.4-12.1) 2.4 (0.6-9.2) 1.7 (0.4-7.1) 2.8 (0.9-9.0) 2.3 (0.7-7.7)

Educational level

None or Elementary 1.1 (0.5-2.9) 0.6 (0.2-1.8) 4.9 (2.4-9.9) 4.5 (2.0-10.1) 0.2 (0.1-0.5) 0.2 (0.07-0.9) 1.6 (0.5-5.1) 3.8 (1.1-14.3) < baccalaureate 0.9 (0.4-2.3) 0.7 (0.2-1.9) 2.4 (1.2-4.9) 2.4 (1.1-5.2) 0.3 (0.1-0.9) 0.3 (0.1-0.9) 2.1 (0.7-5.9) 3.1 (1.0-9.2) Baccalaureate or higher