Charles-Emmanuel Dutoit

Anatoli Stepanov

Johan Van Tol

Maylis Orio

Sylvain Bertaina
email: sylvain.bertaina@im2np.fr

Superlattice Induced by Charge Order in the Organic Spin Chain (TMTTF) 2 X ( X = SbF 6 , AsF 6 and PF 6 ) Revealed by High Field EPR

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

The family of quasi-one-dimensional organic conductors (TMTTF) 2 X is known to have a rich phase diagram with a sequence of competing ground states between spin-Peierls (SP), antiferromagnetic (AF) or superconductor state, depending on the nature of counter anion X or external pressure. More particularly, the centro-symmetric X (X = SbF 6 , AsF 6 and PF 6 ) is metallic at high temperatures, then it is an insulator, and finally, the ground state is either AF for SbF 6 (T N = 8 K) or SP for AsF 6 (T SP = 19K) and PF 6 (T SP = 13 K). The metal-insulator transition was first observed by conductivity [START_REF] Laversanne | Structural and electrical properties of (TMTTF) 2 MF 6 salts (M = P, As, Sb). Role of the anions[END_REF] and microwave measurements 2 , and was attributed to a charge ordering (CO) [START_REF] Shibata | Charge ordering in the one-dimensional extended Hubbard model: Implication to the TMTTF family of organic conductors[END_REF][START_REF] Riera | Influence of the anion potential on the charge ordering in quasi-one-dimensional charge-transfer salts[END_REF] inside the molecule. At T > T CO , the charge (hole) is equality divided between the two TMTTF molecules, then at T < T CO a displacement of charge from one TMTTF to the other induces a 4k f ordering in the direction of the intrastack a axis (spin chain axis). The CO transition was considered structureless because no observation of the superlattice reflection was reported [START_REF] Laversanne | Structural and electrical properties of (TMTTF) 2 MF 6 salts (M = P, As, Sb). Role of the anions[END_REF] . Since it has been proven that the X-ray was responsible for the destruction of the CO transition [START_REF] Foury-Leylekian | Neutron-scattering evidence for a spin-Peierls ground state in (TMTTF) 2 PF 6[END_REF][START_REF] Coulon | Electronic instabilities and irradiation effects in the (TMTTF) 2 X series[END_REF] . Only neutron scattering was able to directly show the displacement of the lattice during the CO transition [START_REF] Foury-Leylekian | Neutron scattering evidence for a lattice displacement at the charge ordering transition of[END_REF] .

The figure 1 shows the elementary cell structure of (TMTTF) 2 X. The electronic properties of this family of compounds are due to a hole shared by the two molecules of TMTTF. At high temperature (T > T CO ), the crystal structure is triclinic with a center of symmetry P 1.

The distances between the sulfur atoms and the nearest counter-anion, are equal, D 1 = D 2 , and the density of hole is the same on each TMTTF [START_REF] Pouget | Structural Aspects of the Bechgaard Salts and Related Compounds[END_REF] . Below T CO , the center of symmetry is removed, the distances D 1 and D 2 are different, leading to a charge disproportion between the two molecules of TMTTF and, as a consequence, to the charge ordering.

The observation of the CO transition in (TMTTF) 2 X has been reported using many techniques. [START_REF] De Souza | Evidence for Lattice Effects at the Charge-Ordering Transition in (TMTTF) 2 X[END_REF] C NMR investigations showed a splitting of lines below T CO , caused by charge rearrangement around 13 C 9,10 . A minimum of the dielectric permittivity at T CO = 156 K for X =SbF 6 , T CO = 103 K for X =AsF 6 and T CO = 67 K for X =PF 6 have been reported by Monceau et al. 11 and the role of the lattice in the CO has been studied by optic [START_REF] Dumm | Charge order in (TMTTF) 2 PF 6 investigated by infrared spectroscopy[END_REF] tool of choice to measure with a high accuracy the g-tensor in such organic crystals where the anisotropy of g-factor is very weak [START_REF] Coulon | Electron spin resonance: a major probe for molecular conductors[END_REF] . In the high temperature phase, the magnetic axesc m and b m are, respectively, parallel and perpendicular to the axis of the TMTTF molecule (Fig. 1c) and are clearly different to the crystallographic axes. The a m axis is perpendicular to the plane formed by the TMTTF and is close to the a axis, making with it an angle of about 3 • . Despite a weak magnetoelectric coupling, the EPR was able to detect the CO transition. Conductive EPR has shown a change of line asymmetry at T CO [START_REF] Coulon | Anisotropic conductivity and charge ordering in (TMTTF) 2 X salts probed by ESR[END_REF] . A new source of line broadening below T CO was reported [START_REF] Furukawa | Deuteration Effect and Possible Origin of the Charge-Ordering Transition of (TMTTF) 2 X[END_REF][START_REF] Nakamura | ESR Investigation of Charge Localized States in (TMTTF) 2 X[END_REF][START_REF] Nakamura | Possible Charge Ordering Patterns of the Paramagnetic Insulating States in (TMTTF) 2 X[END_REF] . On the basis of molecular density-functional theory (DFT) calculations, the rotation of the principal axes of the g-tensor was attributed to the CO transition 20 . This last result is represented in figure 1c andd.

It is important to notice that all these results have assumed or shown a uniform displacement of the anions.

The (TMTTF) 2 X family is also considered as a good prototype of quantum quasi-1D

antiferromagnetic Heisenberg system (QQ1DAFH) with J ∼ 400 K [START_REF] Coulon | Electron spin resonance: a major probe for molecular conductors[END_REF]21,[START_REF] Foury-Leylekian | Inelastic neutron scattering investigation of magnetic excitations in the spin-Peierls ground state of[END_REF] but the magnetic dimensionality remains controversial. Band structure calculations have shown a transfer integral inside the chain of t a ∼ 200 meV and between the chains t b ∼ 40 meV [START_REF] Granier | 4 K crystallographic and electronic structures of (TMTTF) 2 X salts (X -= PF -6 , AsF -6[END_REF][START_REF] Giovannetti | Unraveling the polar state in TMTTF 2 -PF 6 organic crystals[END_REF] and t c ∼ 1 meV 25 in the directions b and c, respectively. From a simple tight binding model [START_REF] Seo | Theoretical aspects of charge ordering in molecular conductors[END_REF] it follows that J a > J b ≫ J c (Using J a ∼ 400 K, we estimate J b ∼ 16 K and J c ∼ 0.01 K

). But recently [START_REF] Yoshimi | Tuning the Magnetic Dimensionality by Charge Ordering in the Molecular TMTTF Salts[END_REF] , it has been shown that such values of the transfer integrals could lead to a more 2D magnetic behavior and the application of QQ1DAFH models 28 might be wrong.

Moreover, on account of the quasi-absence of electronic correlation in the c direction, the majority of theoretical and experimental studies probe the properties in the ab plane only.

In this article, we report a direct observation of superlattice in the direction c, induced by the CO transition. Using high field EPR, we demonstrate that, in addition to the superlattice inside the chain axis, the suppression of the center of symmetry also creates a superlattice between the chains which is resolved when the magnetic field is large enough to decouple the chains in the direction c. The decoupling of the chains allows us to estimate the coupling constant J c .

Experimental details

High-field/high-frequency EPR (HF-EPR) experiments have been carried out using a homemade quasioptical superheterodyn setup developed at NHMFL [START_REF] Van Tol | A quasioptical transient electron spin resonance spectrometer operating at 120 and 240 GHz[END_REF] For clarity the variation of H R from the resonance field at high temperature H 0 is presented.

H 0 =11949.1 mT -0.6 -0.4 -0.2 -0.0 0.2 0.4 X=SbF 6 f=240 GHz H 0 =8530.6 mT H R -H 0 (mT) (a) (b) (c) (d) (e)

Results and discussions

Charge order transition observed by HF-EPR

The figure 2 illustrates the detection of the CO transition by HF-EPR. At high temperature, the EPR line is a Lorentzian with a linewidth of about 3 G close to the one reported at low frequency [START_REF] Nakamura | Possible Charge Ordering Patterns of the Paramagnetic Insulating States in (TMTTF) 2 X[END_REF] . When T < T CO , the line splits in two lines of nearly the same intensity.

Examples of raw spectra are given in Fig. 2.a and Fig. 2.b for X =AsF 6 and PF 6 respectively.

For AsF 6 the splitting was observed at all available frequencies, while for SbF 6 it was observed at 240 GHz and 336 GHz, and for PF 6 at 336 GHz only. The resonance fields of the compounds are reported in Fig. 2.c, Fig. 2.d and Fig. 2.e. The splitting is reversible (returning back to high temperature makes the lines to collapse) and reproducible (many samples from different batches have been used). It is clear that CO is responsible for the splitting of the EPR lines, but the physical interpretation is not trivial. It is known that CO removes the center of symmetry 30 and reduces the group from P 1 to P1. The electronic density is no more equivalent between the two TMTTF molecules, which leads to two [START_REF] De Souza | Evidence for Lattice Effects at the Charge-Ordering Transition in (TMTTF) 2 X[END_REF] C NMR signals [START_REF] Chow | Charge Ordering in the TMTTF Family of Molecular Conductors[END_REF] .

However, the effect is different in the case of EPR, where only one electron spin is shared by the two TMTTF molecules, and thus only one signal is expected. Now, following the results of Riera et Poilblanc 4 , let's assume that the displacement of the anions is not fully uniform and has a small modulation in space. In Fig. 3 we present two simple models. A and B are 2 representations of the unit cell in the CO state. In B the displacement of the anions X is a bit stronger than in A. Consequently, in B the electron-rich TMTTF molecule is a bit richer than in A. Although the charge disproportion has no direct effect on the spin, the counter-anion displacement has a similar effect on the g-tensor. In triclinic symmetry, the the orientation of g-tensor axes is not fixed by the symmetry and the displacement of X induces a rotation of the tensor principal axes 20 . A similar effect has been reported for other low-symmetry systems [START_REF] Van Rens | Single-Crystal EPR Study of a Cu(II)dialkyldiselenocarbamate. An Example of Noncoinciding Principal Axes[END_REF][START_REF] Pilbrow | Rotation of tensor axes in the EPR of Cr 3+ in monoclinic sites[END_REF] . In the configuration B the displacement is stronger than in A, which leads to a more significant rotation of the g-tensor of the configuration B than in A.

Exchange splitting

The ability to resolve the two EPR lines coming from the A and B configurations depends on relative value of the mismatch of the Zeeman energies for the two non-equivalent spins, ∆gµ B H, and on the Heisenberg exchange interaction in the direction i, k B J i . When the magnetic field is smaller than the exchange interaction, the signals from the two non equivalent sites are merged into one line due to fast fluctuation, as predicted by the theory of exchange narrowing [START_REF] Anderson | Exchange Narrowing in Paramagnetic Resonance[END_REF][START_REF] Kubo | A General Theory of Magnetic Resonance Absorption[END_REF] . On the contrary, the lines from the two non equivalent magnetic sites are split [START_REF] Hennessy | Effect of Interchain Coupling on Electron-Spin Resonance in Nearly One-Dimensional Systems[END_REF] in the case of strong magnetic field:

∆gµ B H > k B J i . ( 1
)
Here ∆g is the difference of the g factors of the two non-equivalent sites; it is a maximum at 45 • between b m and c m . This condition, mathematically proven by P.W. Anderson [START_REF] Anderson | A Mathematical Model for the Narrowing of Spectral Lines by Exchange or Motion[END_REF] , was used to directly estimate the interchain coupling in the quantum spin chain CuGeO 3 37 . The intrachain exchange interaction of (TMTTF) 2 X (∼400 K) is too large and all the chains in the configuration (π, π) in Fig. 3 should have the same resonance field (only one line is expected).

In the configuration (0, π), two kinds of chains exist : full A and full B. Compared to Ref. [START_REF] Riera | Influence of the anion potential on the charge ordering in quasi-one-dimensional charge-transfer salts[END_REF] , the configuration (π, π) is the scenario (c), (π, 0) is the scenario (d), (0, 0) is the scenario (a)

(and is the standard picture to explain the CO transition), (0, π) is not mentioned. Since we can resolve two lines, we are able to decouple the signals from two different chains. Thus we are able to give an upper limit for the smallest interchain coupling J c .

To our knowledge, this is the first experimental estimation of the interaction between chains of (TMTTF) 2 X in the direction c and this result has remarkable implications for the magnetic properties.

It confirms the quasi absence of electronic correlation in the c direction. Band structure calculations have managed to estimate the transfer integral in the a (chain axis) and b directions [START_REF] Granier | 4 K crystallographic and electronic structures of (TMTTF) 2 X salts (X -= PF -6 , AsF -6[END_REF][START_REF] Giovannetti | Unraveling the polar state in TMTTF 2 -PF 6 organic crystals[END_REF] but only one paper, so far, has reported a theoretical prediction in the c direction t c ∼ 1 meV, which leads in the tight binding model to J c = (t c /t a ) 2 J ∼ 10 mK) 25 one order of magnitude higher than what we get but not so far away from theoretical predictions using mean field models. Indeed, the magnetic dimensionality has to be carefully considered.

Although the (TMTTF) 2 X family is a QQ1DAFH system, the large difference between J b and J c makes some models fail. One remarkable example is the estimation of the interchain coupling J ′ from the Néel order temperature [START_REF] Yasuda | Néel Temperature of Quasi-Low-Dimensional Heisenberg Antiferromagnets[END_REF][START_REF] Irkhin | Calculation of Neel temperature for S=1/2 Heisenberg quasi-one-dimensional antiferromagnets[END_REF] :

J ′ = 1.073T 1D N ln 2.6J T 1D N + 1 2 ln ln 2.6J T 1D N ( 2
)
which gives for (TMTTF) 2 SbF 6 , J ′ ∼ 2 K, three orders of magnitude higher than the value we found. The reason comes from the assumptions made to develop the standard model by Irkin and Katanin 38 then Yasuda et al. : The chain is coupled to the nearest neighbor chains by the same interaction. This is not the case for (TMTTF) 2 SbF 6 . More surprisingly, Yoshimi et al. [START_REF] Yoshimi | Tuning the Magnetic Dimensionality by Charge Ordering in the Molecular TMTTF Salts[END_REF] have shown that in the CO state the inter-site Coulomb repulsion could lead to an exchange interaction of the same order in direction a and b. However, this last result is not confirmed by our experimental results, since a QQ2DAFH should have a much higher Néel temperature, even with the weak J c we report. Using the equation connecting the Néel temperature and the interlayer coupling J ′ :

T 2D N = 0.732πJ 2.43 -ln J ′ J -ln T 2D N J ( 3
)
with J = 400 K, J ′ = 1 mK, we find T 2D N = 54 K, while T 1D N = 3 mK from eq.( 2). In both cases, the model fails to describe (TMTTF) 2 SbF 6 . Our results should help the extension of the 1D and 2D models connecting T N and the exchange couplings [START_REF] Yasuda | Néel Temperature of Quasi-Low-Dimensional Heisenberg Antiferromagnets[END_REF][START_REF] Irkhin | Calculation of Neel temperature for S=1/2 Heisenberg quasi-one-dimensional antiferromagnets[END_REF] to the cases of intermediate dimensionality, J a > J b ≫ J c .

Charge ordering and the superlattice in the c direction.

In order to understand the role of the CO transition on the appearance of a magnetic superlattice in direction c, we performed EPR measurements for several orientations in the plane perpendicular to the chain axis. Fig. 4 shows the variation of the resonance field as a function of the angle θ in (TMTTF) 2 AsF 6 for f = 336 GHz above (black diamonds) and below (red circles and blue squares) T CO . θ = 0 • corresponds to the maximum of the resonance field at T = 110 K. At T < T CO the splitting of the EPR line is accompanied by a rotation of the g-tensor. The rotation is not the same for the two lines. In the case of line 1, we found a rotation through 3 • , whereas for line 2 the angle of rotation is 11 • . The rotation of the principal axes of the g-tensor was observed at low field in non-resolved EPR spectra and explained by the displacement of the anions X 20 . The modulation of the stain field in c direction induces two kinds of anion displacements leading to 2 different rotations of the g-tensor which can be resolved at high field.

The relative rotation of the g-tensor (∆θ) of two adjacent chains was also assumed to explain the EPR line broadening for T < T CO at lower frequencies [START_REF] Yasin | Broken magnetic symmetry due to charge-order ferroelectricity discovered in (TMTTF) 2 X salts by multifrequency ESR[END_REF] . However neither the rotation axis nor the amplitude of the rotation is coherent with our results. The reason is that the authors used the anisotropic Zeeman effect (AZE) [START_REF] Pilawa | Anisotropy of the electron spin-resonance linewidth of CuGeO 3[END_REF] in order to extract quantitative information from the EPR linewidth . Unfortunately, this model needs the interchain coupling J ′ and in the absence of a reliable value, the authors estimated J ′ using 28 , which, as we have shown previously, is irrelevant in the case of (TMTTF) 2 X salts. Using J ′ = 1.1 K and an enhanced linewidth of 1.5 G they found a large relative rotation of ∆θ = ±32 • about the b axis. Now let's approach the problem from the other side by using our direct measurement of ∆θ = ±4 • about the a axis which yields ∆g ≈ 2.10 -4 , and the experimental results obtained in W band from [START_REF] Yasin | Broken magnetic symmetry due to charge-order ferroelectricity discovered in (TMTTF) 2 X salts by multifrequency ESR[END_REF] (∆H = 1.5 G and H 0 = 3.36 T). Applying the the AZE model :

J ′ = π/8H 2 0 |∆g| 2 /(g e ∆H) (4)
Figure 4: (color online). Resonance field of (TMTTF) 2 AsF 6 recorded in the plane perpendicular to the chain axis. The black diamonds represent the line at T = 110 K (above T CO ) while the red circles and the blue squares are the two lines observed at T = 60 K (below T CO ). The lines are the best fits. The insets represent the g-tensor axes in the plane perpendicular to the chain axis. At T > T CO all the chains are equivalent. At T < T CO two inequivalent chains have different orientations of their g-tensors.

we find J ′ ∼ 0.9 mK which is coherent with our values of J c (Table 1).

The figure 5 demonstrates how EPR can probe the CO transition. Fig. 5a: the TMTTF (brown) molecules are stacked in the a direction and form the chain axis. They are separated by the counter anions X (PF 6 , AsF 6 , SbF 6 ) (green). Due to the low triclinic symmetry of the system, the crystallographic axes and the magnetic axes are different. For T > T CO the 2 TMTTF molecules are equivalent and the magnetic axes c m and b m coincide with the symmetry axes of the TMTTF. At the CO phase transition (Fig. 5b), the symmetry is reduced by removing the centers of inversion, the counter anion X cages are shifted and the charge balance on the two TMTTF is broken. One TMTTF (blue) has a higher charge density than the second one (red). Since the g-tensor is sensitive to the electronic change 20 , it turns about the a axis (as shown by low filed EPR 20,41 ). The vertical sinusoidal curve represents the change of the electronic density. Fig. 5c shows how the high-field/high-frequency EPR revealed that the CO is not uniform and a CO in the c direction is also induced. In the chains with light (dark) red and blue TMTTF the charge displacement is less (more) pronounced.

As a consequence, the rotation of the g-tensor is less (more) significant, resulting in the line 1 (line 2).

Finally, let's discuss why the superlattice in the c direction was not detected by other techniques. In the introduction to this paper we have presented several techniques which have permitted he observation of the CO. However, only few of them such as [START_REF] De Souza | Evidence for Lattice Effects at the Charge-Ordering Transition in (TMTTF) 2 X[END_REF] C NMR or neutron scattering, have the spectral or spatial resolution to detect the interchain CO modulation. In the case of 13 C NMR, the main CO was detected by observing the splitting of [START_REF] De Souza | Evidence for Lattice Effects at the Charge-Ordering Transition in (TMTTF) 2 X[END_REF] C NMR lines but, as seen in Fig. 2 the rotation of the g-tensor axes. In higher symmetry, this effect would not exist. (ii) The very small linewidth of the organic salts (∆H ∼ 1 G) allows to resolve the two EPR lines.

(iii) The two EPR lines are not collapsed by the interchain exchange interaction because electronic correlations in the c direction were nearly absent and the magnetic field was high (exchange splitting regime [START_REF] Anderson | A Mathematical Model for the Narrowing of Spectral Lines by Exchange or Motion[END_REF] ).

In conclusion, the CO transition observed in (TMTTF) 2 X (X =SbF 6 , AsF 6 , PF 6 ) is accompanied by a displacement of the counter anions X. This displacement was observed by neutron 7 , NMR 9 and dilatometry 11 but was considered uniform 13 all over the crystal.

Using high field/frequency EPR, we have seen the signal of two non-equivalent spin chains.

We have interpreted this result as a modulation of the displacement of X in the direction c leading to two orientations of the g-tensor and consequently to two EPR lines.

The line splitting allows us to estimate the exchange coupling in the c direction (J c ∼ 1 mK). Using this value, we have shown that neither the 1D model (equation ( 2)) nor the 2D one ( equation ( 3)) can describe (TMTTF) 2 SbF 6 . We believe that this result should stimulate the development of a theory linking T N and the inter chain couplings in the case

J a > J b ≫ J c .

Figure 1 :

 1 Figure1: (color online). Crystallographic structure of (TMTTF) 2 X. Two molecules of TMTTF (in the center of the cell) share one hole given by one of the eight counter anions X -(X =SbF 6 , AsF 6 or PF 6 ).

.

  The spectrometer operates at 120 GHz, 240 GHz and 336 GHz and at temperature from RT down to 2 K. The inhomogeneity of the field (crucial in our results) is less than 0.1 G across the volume of the samples. The presence of modulation coils allows us to record the first derivative of both the absorption and dispersion signals. The single crystals of (TMTTF) 2 X with X = SbF 6 , AsF 6 and PF 6 have the shape of a needle with a typical size of 50 × 100 × 700 µm[START_REF] Shibata | Charge ordering in the one-dimensional extended Hubbard model: Implication to the TMTTF family of organic conductors[END_REF] , small enough to avoid polariton reflections inside the sample. The angular dependence of the EPR was measured by using a goniometer which rotates the sample in the plane perpendicular to the a axis. Due to the triclinic symmetry, the magnetic and crystallographic axes are different. The magnetic axes b m and c m correspond to the minimum and maximum of the resonance field respectively. The temperature dependence of the EPR has been measured by applying the field at 45 • between b m and c m . The HF-EPR spectra were recorded for the three systems and for the three available frequencies.

Figure 2 :

 2 Figure 2: (color online) HF-EPR recorded at temperature above and below T CO for the magnetic field at 45 • between the magnetic axis b m and c m . (a) Derivative absorption signals of (TMTTF) 2 AsF 6 at f=240 GHz when the temperature decreases.(b) Derivative dispersion signals of (TMTTF) 2 PF 6 at f=336 GHz when the temperature decreases. The red circle and blue square are the resonance fields H R and are reported as a function of temperature in (c), (d), (e) for X = SbF 6 , AsF 6 and PF 6 respectively. The vertical dashed lines represent T CO .For clarity the variation of H R from the resonance field at high temperature H 0 is presented.

Figure 3 :

 3 Figure 3: (color online). Schematic representation of CO superlattice inspired from the original work of Riera and Poilblanc 4 . A and B are unit cells of (TMTTF) 2 X in the CO configuration (ie : with charge density displacement ) but B has a stronger charge displacement than A. The vertical dashed lines represent the chain axis. The (π, π) configuration alternates A and B in both directions whereas (0, π) has a uniform A or B inside the chain but alternates between the chains.

  and 4 of Ref[START_REF] Chow | Charge Ordering in the TMTTF Family of Molecular Conductors[END_REF] , this splitting is very small. It is not surprising that a modulation of the CO would lead to a second separation of each line too tiny to be resolved. The measurement of the displacement of the counter anions responsible for the CO was a challenging neutron scattering experiment[START_REF] Foury-Leylekian | Neutron scattering evidence for a lattice displacement at the charge ordering transition of[END_REF] . The modulation of the displacement would just be included in the error bars. In our case the observation of the CO modulation was possible because of : (i) The low P 1 symmetry which allows

Figure 5 :

 5 Figure5: (color online). Schematic representation of the structural and magnetic properties of (TMTTF) 2 X. The left column represents 4 unit cells in the plane perpendicular to the chain axis a while the right side is 4 unit cells in the plane ac. (a) At T > T CO , the charge density is equally distributed between the TMTTF molecules(brown). The principal axis of th g-tensor (pink) are oriented along the TMTTF molecular axes. All the unit cells are magnetically equivalent. (b) At T < T CO the charge density of the two TMTTF molecules is no more balanced and the g-tensor axes rotate about the a axis 20 . (c) The high-field/highfrequency EPR experiments reveal that this rotation is not uniform due to modulation of the charge ordering. The charge unbalance is stronger in one stack (dark blue and red) than in the second (light blue and red) leading to two different rotations of the g-tensor axes and thus to two EPR lines.15

Table 1 : Estimation of the interchain coupling J c . ∆g is the difference of the g factors of the two non-equivalent sites for

 1 θ = 45 • .

The frequencies/fields correspond to the threshold where the splitting has been resolved. Using (1) we estimate the upper limit of the interchain coupling.

	X =	∆g	minimum frequency/(field) to resolve the splitting	upper limit J c
	AsF 6 12.10 -5	120 GHz (4.28 T)	5G (∼ 0.7 mK)
	SbF 6 9.10 -5	240 GHz (8.56 T)	8G (∼ 1 mK)
	PF 6	7.10 -5	336 GHz (12 T)	8G (∼ 1 mK)

Acknowledgement.

We acknowledge M. Dressel for providing the sample. We thank M. Kuzmin and S. Todo for stimulating discussions. This work was supported by NSF Grant No. DMR-1206267, CNRS-PICS CoDyLow and CNRS's infrastructure of research RENARD (FR3443) for EPR facilities. The NHMFL is supported by the Cooperative Agreement Grant No. DMR-1157490 and the State of Florida.


Sylvain Le Corff

Matthieu Lerasle

Elodie Vernet

A Bayesian nonparametric approach for generalized Bradley-Terry models in random environment

. This paper provides three contributions on a Bayesian nonparametric approach to solve this problem. First, we establish contraction rates of the posterior distribution. We also propose a Markov Chain Monte Carlo algorithm to approximately sample from this posterior distribution inspired from a recent Bayesian nonparametric method for hidden Markov models. Finally, the performance of this algorithm are appreciated by comparing predictions on the issue of a championship based on the actual values of the teams and those obtained by sampling from the estimated posterior distribution.

Introduction

This paper considers a class of models to deal with pairwise comparison of individuals. The outcome of all the successive comparisons is described by an ordering after all the contestants have met once. A potentially large number of participants are ranked using scores which aggregate the outcomes of their meetings. The objective is to predict this ranking when only a small number of pairwise comparisons has been observed. In generalized Bradley-Terry models, each player i 1 is associated with an unknown real parameter V i characterizing his strength or ability. When contestants i and j face each other, the outcome of their meeting is a random variable taking values in a discrete set X . For all x ∈ X , this random variable equals x with probability K(x, V i , V j ), where the function K is known. The most famous example of such a function K is the Bradley-Terry model [START_REF] Bradley | Rank analysis of incomplete block designs: I. the method of pair comparisons[END_REF][START_REF] Zermelo | Die Berechnung der Turnier-Ergebnisse als ein Maximumproblem der Wahrscheinlichkeitsrechnung[END_REF] in which

X = {0, 1}, K(x, V i , V j ) = V x i V 1-x j V i + V j . (1)
If x = 1, i has beaten j, otherwise, j has beaten i. These models have emerged as key tools to describe pairwise comparisons and have been applied to chess ranking [START_REF] Joe | Extended use of paired comparison models, with application to chess rankings[END_REF], sports [START_REF] Sire | Understanding baseball team standings and streaks[END_REF], animal behaviour [START_REF] Whiting | Ultraviolet signals ultra-aggression in a lizard[END_REF]. The maximum likelihood estimator (MLE) of the strengths V i was studied theoretically in [START_REF] Simons | Asymptotics when the number of parameters tends to infinity in the Bradley-Terry model for paired comparisons[END_REF] when each contestant has faced all the others once. This result has led to interesting developments in computational statistics to design efficient algorithms to approximate the MLE [START_REF] Caron | Efficient Bayesian inference for generalized Bradley-Terry models[END_REF][START_REF] Hunter | MM algorithms for generalized Bradley-Terry models[END_REF]. The assumption that each contestant has faced all the others is restrictive in some applications and there has been several attempt to weaken it [START_REF] Diel | Learning latent structure of large random graphs[END_REF][START_REF] Yan | Sparse paired comparisons in the Bradley-Terry model[END_REF]. However, it cannot be completely relaxed as the MLE only exists in the Bradley-Terry model if there exists a path between each pair of players in the oriented graph where an edge is drawn from i to j if i has beaten j [START_REF] Zermelo | Die Berechnung der Turnier-Ergebnisse als ein Maximumproblem der Wahrscheinlichkeitsrechnung[END_REF]. When only a few contests have been observed, it is very likely that the previous condition is not met. Studying the scores in the American baseball league, [START_REF] Sire | Understanding baseball team standings and streaks[END_REF] introduced a simple model where the parameters V i are independent and identically distributed (i.i.d.) with common uniform distribution on [α, 1] for some parameter α > 0 that has to be estimated. The authors obtained striking results for the prediction of consecutive-game team winning and losing streaks for instance. This motivated in [START_REF] Chetrite | The number of potential winners in bradley-terry models in random environment[END_REF] the introduction of the Bradley-Terry model in random environment where the function K is given by (1) and the strengths (V i ) i 1 are assumed i.i.d. with common distribution π on (0, +∞). The authors proved in [START_REF] Chetrite | The number of potential winners in bradley-terry models in random environment[END_REF] that in the case of general Bradley-Terry models in random environment, the winner of a championship is the one with maximal strength if and only if the tail of π is sufficiently convex. In other words, interesting predictions regarding the outcome of a tournament can be inferred from π. This is why [START_REF] Diel | Learning latent structure of large random graphs[END_REF] considered the problem of estimating the distribution of the (V i ) i 1 in generalized Bradley-Terry models in random environment, showing that the MLE of this distribution can be defined even when each player has only been involved in 2 contests. They also studied the MLE of π when players meet according to the round-robin scheduling, a widely spread method employed by tournament schedulers. The cornerstone of the work presented in [START_REF] Diel | Learning latent structure of large random graphs[END_REF] is that risk bounds for the MLE of π follow from the analysis of the likelihood of the following graphical model [ The details of the link between the graphical model in Figure 1 and (generalized) Bradley-Terry models can be found in [7, Section 2]. Roughly speaking, in Figure 1, each Vi , 1 i n+1, gathers a group (V j ) j∈Gi of contestants in the Bradley-Terry model and each Xi , 1 i n, gathers the outcomes of meetings between contestants j and k when both belong to G i or when j ∈ G i and k ∈ G i+1 . In this representation, it is shown in [START_REF] Diel | Learning latent structure of large random graphs[END_REF] that when meetings are scheduled according to the round-robin algorithm the Markov chain ( Xi , Vi+1 ) 1 i n is stationary.

The contribution of this paper can be summarized as follows. First, in a Bayesian formulation of the problem considered in [START_REF] Diel | Learning latent structure of large random graphs[END_REF], Section 3 provides the first nonparametric posterior concentration rates for the estimation of the distribution of the hidden variables (V i ) 1 i n+1 associated with the graphical model given in Figure 1. Then, Section 4 introduces an algorithm to simulate this posterior distribution. In particular, this algorithm allows to compute an estimator of π and to observe the actual performance of our estimation strategy. These practical aspects are illustrated in a simulation study inspired by data from season 2017-2018 of the French Ligue 1 in Section 4.

In the case of i.i.d. observations, posterior concentration rates have been obtained for a large class of prior distributions and target densities, see [START_REF] Kruijer | Adaptive Bayesian density estimation with location-scale mixtures[END_REF][START_REF] Shen | Adaptive Bayesian multivariate density estimation with Dirichlet mixtures[END_REF] for Gaussian mixtures and [START_REF]Convergence rates for density estimation with Bernstein polynomials[END_REF][START_REF] Rousseau | Rates of convergence for the posterior distributions of mixtures of betas and adaptive nonparametric estimation of the density[END_REF] for Beta mixtures. Overviews on recent developments of the nonparametric Bayesian theory can be found in [START_REF] Rousseau | On the frequentist properties of Bayesian nonparametric methods[END_REF][START_REF] Ghosal | Fundamentals of nonparametric Bayesian inference[END_REF] and the references therein. Posterior concentration rates were established for partially observed dependent data by [START_REF] Vernet | Non parametric hidden markov models with finite state space: Posterior concentration rates[END_REF] following the seminal paper [START_REF] Ghosal | Convergence rates of posterior distributions for non-i.i.d. observations[END_REF]. In [START_REF] Vernet | Non parametric hidden markov models with finite state space: Posterior concentration rates[END_REF], the observations arise from a discrete state space hidden Markov model and minimax posterior concentration rates are obtained for the finite-dimensional distributions of the observations. In this paper, posterior concentration rates are obtained for the unknown distribution of latent variables which take values in a continuous state space. Following the approach of [START_REF] Ghosal | Convergence rates of posterior distributions for non-i.i.d. observations[END_REF] (see also [START_REF] Rousseau | On the frequentist properties of Bayesian nonparametric methods[END_REF]) this requires in particular to design exponentially consistent tests by using concentration inequalities for Bradley-Terry models.

An important difference with the frequentist analysis performed in [START_REF] Diel | Learning latent structure of large random graphs[END_REF] is that these inequalities have to be uniform in π to bound the power of these tests. The main results given in Section 3 require therefore a substantial extension of the technical tools used in [START_REF] Diel | Learning latent structure of large random graphs[END_REF] even if the underlying strategy is similar.

The Bayesian nonparametric procedure introduced in this paper to sample from the posterior distribution of π is inspired by the mixture of Dirichlet processes and the data augmentation scheme presented in [START_REF] Yau | Bayesian non-parametric hidden Markov models with applications in genomics[END_REF]. The algorithm proposed by [START_REF] Yau | Bayesian non-parametric hidden Markov models with applications in genomics[END_REF] is a Markov Chain Monte Carlo (MCMC) algorithm applied for discrete hidden Markov models with conditional distribution specified as a mixture model in which a base measure is mixed with respect to a Dirichlet process prior (DPP). In Section 4, we propose a mixture of Dirichlet processes formulation where the unknown distribution π is defined as a mixture model where Gaussian distributions are mixed with respect to a DPP. The joint posterior distribution of the parameters of both the DPP and the hidden variables (V i ) 1 i n+1 is sampled from by adapting the block Gibbs sampling mechanism of [START_REF] Yau | Bayesian non-parametric hidden Markov models with applications in genomics[END_REF] to our setting. The main challenge is to sample from the posterior distribution of the states (V i ) 1 i n+1 . Contrary to the discrete case setting considered in [START_REF] Yau | Bayesian non-parametric hidden Markov models with applications in genomics[END_REF], exact sampling from this distribution is not possible here and a Sequential Monte Carlo algorithm is used instead. Section 4 presents the performance of the overall procedure when the Forward Filtering Backward Simulation algorithm proposed by [START_REF] Godsill | Monte Carlo smoothing for non-linear time series[END_REF] (see also [START_REF] Douc | Sequential Monte Carlo smoothing for general state space hidden Markov models[END_REF]) to perform this step is used.

The remaining of the paper is divided as follows. Section 2 displays the formal setting of the paper and Section 3 the main results: posterior concentration rates are established under standard assumptions on the prior distribution. Examples of priors satisfying these assumptions are also given. Section 4 presents the numerical experiments. The MCMC algorithm to sample from the posterior distribution is introduced and an application to a simulated dataset inspired from the results of the French soccer championship is also discussed.

Setting

Let n be a positive integer and (V i ) 1 i n+1 denote i.i.d. random variables taking values in a measurable set (V, B(V)). For all 1 i n, the observation X i takes values in a discrete set X and conditionally on (V i ) 1 i n+1 the random variables (X i ) 1 i n are independent. The conditional distribution of X i given (V k ) 1 k n+1 depends on V i and V i+1 only and is denoted by

x → K(x, V i , V i+1 ), where K : X × V × V → [0, 1]
is known. Let Π be a set of probability measures on (V, B(V)). For all π ∈ Π, note that when the probability distribution of each V i , 1 i n + 1, is π, then the joint sequence (X i , V i+1 ) 1 i n is a stationary Markov chain with transition kernel P : X × V → P(X ) ⊗ B(V) given, for all (x, v) ∈ X × V and all (x , A) ∈ X × B(V), by

P π (x, v; x , A) = 1 A (v )π(dv )K(x , v, v ) , (2)
where 1 A denotes the indicator function of the set A. This Markov chain may be extended to a stationary process (X i , V i+1 ) i∈Z with the same transition kernel P π . In the following, we use the shorthand notation X = (X i ) i∈Z and V = (V i ) i∈Z . The joint law of this extended joint process is denoted by P π and the associated expectation is written E π . For all 1 i n, and all π ∈ Π,

P π (X i |V, X 1:i-1 ) = P π (X i |V i , V i+1 ) = K (X i , V i , V i+1 ) ,
where for any sequence (a ) ∈Z ,

a u:v = (a u , . . . , a v ) if u ≤ v and a u:v = ∅ if u > v.
These conditional dependences are represented in the graphical model given in Figure 2. In the Bayesian setting of this paper, the set of distributions Π is equipped with a sigma-algebra A and a prior distribution µ. The posterior distribution of any π ∈ Π given the observations is defined, for all A ∈ A, by

V 1 X 1 V 2 X 2 V 3 V n . . . X n V n+1
µ (π ∈ A|X 1:n ) = n i=1 K(X i , v i , v i+1 )1 A (π)π ⊗n+1 (dv 1:n+1 )µ(dπ) n i=1 K(X i , v i , v i+1 )π ⊗n+1 (dv 1:n+1 )µ(dπ) .
In the following, π denotes the true value of the unknown density. This paper focuses on the frequentist properties of the posterior distribution µ (•|X 1:n ). The aim is to establish rates at which the posterior distribution concentrates around π for a well suited loss function. In the following, let γ be a σ-finite distribution on (V, B(V)) and S + be the set of probability densities with respect to γ. S + is equipped with the topology induced by the L 1 -norm and the corresponding Borel σ-field is written B(S + ).

Posterior concentration rates 3.1 Main results

Consider the following assumption.

H1 There exists 0 < ν < 1 such that for all x ∈ X and all v, w ∈ V, K(x, v, w) ≥ ν.

Under H1, the transition kernel of the Markov chain (V i+1 , X i ) i∈Z satisfies, for all i ∈ Z and all A ∈ B(V),

P π (X i-1 , V i ; X i , A) = 1 A (v i+1 )π(dv i+1 )K(X i , V i , v i+1 ) νπ(A) . (3)
This uniform lower bound ensures that the joint Markov chain (V i+1 , X i ) i∈Z is uniformly ergodic and that the whole space V × X is a small set. Theorem 3 establishes the posterior concentration rates under P π , it is derived from two important results. Following [START_REF] Rousseau | On the frequentist properties of Bayesian nonparametric methods[END_REF] and the references therein, Kullback-Leibler type conditions are first required. Kullback-Leibler controls are obtained in Proposition 1. For any π ∈ Π and any x 1:n ∈ X n , let n (π, x 1:n ) be the loglikelihood of the observations

x 1:n , n (π, x 1:n ) = log n i=1 K(x i , v i , v i+1 )π ⊗n+1 (dv 1:n+1 ) . (4)
Let • tv be the total variation distance between probability measures: for all π 1 , π 2 ∈ Π,

π 1 -π 2 tv = sup A∈B(V) |π 1 (A) -π 2 (A)| .
Proposition 1. Assume that H1 holds. Then, there exists a constant c ν > 0 such that for all distributions π, π ∈ Π and all n 1,

E π [ n (π, X 1:n ) -n (π , X 1:n )] c ν π -π 2 tv n (5)
and

Var π [ n (π, X 1:n ) -n (π , X 1:n )] c ν π -π 2 tv log 2 ( π -π tv ) n . (6)
Proof. The proof is postponed to Section A.1.

To prove the existence of exponentially consistent tests, required in [START_REF] Rousseau | On the frequentist properties of Bayesian nonparametric methods[END_REF] for example, we use the concentration inequality obained in Proposition 2. Following [11, Proposition 1] (see also [START_REF] Diel | Learning latent structure of large random graphs[END_REF]), it is established in Lemma 7 that, under H1, for all π ∈ Π, there exists a function π such that P π -a.s. and in L 1 (P π ),

1 n n (π, X 1:n ) -→ n→∞ L π (π) = E π [ π (X)] . (7)
The quantity π (X) is the P π -a.s. limit of the sequence (log P π (X 0 |X -:-1 )) 1 . For all subset Π ⊂ Π compact for the topology induced by • tv , let N( Π, • tv , ε) be the minimal number of balls of radius ε necessary to cover Π. For all π, π ∈ Π and all n 1, define

∆ n π,π (X 1:n ) = 1 n n (π , X 1:n ) -L π (π ) . (8)
Proposition 2. Assume that H1 holds. Then, there exists a constant c ν > 0 such that for all compact subset Π ⊂ Π for the topology induced by • tv , all π ∈ Π, all ε > 0 and all t > 0,

P π   sup π ,π ∈ Π ∆ n π,π (X 1:n ) -∆ n π,π (X 1:n ) c ν   ε + log 2N( Π, • tv , ε) + t √ n     e -t 2 .
Proof. The proof is postponed to Section A.2.

For all ε > 0, define

S (ε) = {π : π -π tv ε} . (9)
Consider the following assumption.

H2 -(c 1 , c 2 ) There exist a sequence (Π n ) n 1 of subsets of Π and two sequences (

ε n ) n 1 and (ε n ) n 1 such that (nε 2 n ) -1 log 2 (ε n ) = o(1), εn ≤ ε n satisfying log N(Π n , • tv , ε n ) ≤ nε 2 n , µ (S (ε n )) e -c1nε 2 n and µ (Π c n ) e -c2nε 2 n .
As in [START_REF] Diel | Learning latent structure of large random graphs[END_REF], using that for all π ∈ Π, π ∈ Argmax L π , define the risk function on Π as

R π : π → L π (π ) -L π (π)
and for all ε > 0 and all constant c > 0,

B c, (ε) = {π ∈ Π : R π (π) cε} .
Under assumption H2, posterior concentraction rates are derived with respect to the risk function R π .

Theorem 3. Assume that H1 and H2 -(

c 1 , c 2 ) hold for constants c 1 , c 2 > 0 such that c 2 > c ν + c 1 where c ν is defined in Proposition 1. For any positive sequence (α n ) n 1 such that α -1 n = O(e cαnε 2 n ) with 0 < c α < c 2 -c ν -c 1 and any c > 2(1 + √ 2) 1 + √ c 2 c ν , P π µ B c c, (ε n ) X 1:n > α n = o(1) .
Proof. The proof is postponed to Section A.3.

Theorem 3 establishes convergence rates of the posterior distribution for the Kullback-Leibler risk R π . The Kullback-Leibler divergence between two probability distributions usually scales as the square norm between these probabilities, so that a scaling in ε 2 n could have been expected in Theorem 3. The reason is that we use the approach of [START_REF] Diel | Learning latent structure of large random graphs[END_REF] that builds on the analysis of empirical risk minimizers by Vapnik [START_REF] Vapnik | Statistical learning theory[END_REF] to bound the risk which results in "slow" rates of convergence of the estimators. "Fast" rates (see [START_REF] Alexandre | Optimal aggregation of classifiers in statistical learning[END_REF] for a discussion on fast and slow rates in learning) in ε 2 n could be derived under margin type assumptions [START_REF] Mammen | Smooth discrimination analysis[END_REF][START_REF] Koltchinskii | Oracle inequalities in empirical risk minimization and sparse recovery problems[END_REF]. Proving that such margin conditions are satisfied in our framework is an interesting question that goes beyond the scope of this paper.

Examples of prior distributions

This section presents classical examples of prior distributions satisfying the assumptions of Theorem 3.

Dirichlet process mixtures of Gaussian distributions Dirichlet process mixtures of Gaussian distributions are commonly used to model densities on R. In [START_REF] Shen | Adaptive Bayesian multivariate density estimation with Dirichlet mixtures[END_REF] and [START_REF] Kruijer | Adaptive Bayesian density estimation with location-scale mixtures[END_REF], this prior distribution is used to obtain adaptive posterior concentration rates for i.i.d. observations. Let G denote a probability measure on R, with positive density on R with respect to the Lebesgue measure and let Π σ denote a probability measure on (0, +∞). These measures are assumed to satisfy the following tail assumptions.

H3 -(κ) There exist (b

i ) i∈{0,1} , x 0 > 0 such that ∀x > x 0 , 1 -G([-x, x]) b 0 e -b1x b 1 , ∀x < x -1 0 , Π σ ([0, x]) b 0 e -b1x -b 1 , ∀x > x 0 , Π σ ([x, +∞)) b 0 x -b1 , ∀s > 0, ∀t ∈ (0, 1), Π σ ((s, s(1 + t))) b 1 s -b0 t b0 e -b0s -κ/2 .
Let a > 0, DP(aG) be the Dirichlet process with base measure aG, φ σ be the Gaussian density function with mean 0 and variance σ on R and assume that under the prior distribution

π = φ σ (. -µ)dP (µ), (P, σ) ∼ DP(aG) ⊗ Π σ . (10)
Let β > 0, τ 0, L a non-negative function on R and let C(β, L, τ ) be the class of locally-Hölder functions with regularity β:

C(β, L, τ ) = f : R → R : ∀(x, y) ∈ R 2 , f ( β ) (x) -f ( β ) (y) L(y)e τ |x-y| 2 |x -y| β-β ,
where β denotes the largest integer smaller than β and, for any integer k, f (k) denotes the k-th derivative of f . Consider also the following assumption.

H4 -(β, L)

There exist positive constants δ, x m , b 4 , b 5 and τ such that

∀k β , |π (k) (x)| π (x) (2β+δ)/k π (dx) < ∞ , L(x) π (x) (2β+δ)/β π (dx) < ∞ , ∀|x| > x m , π (
all t > t 0 > (d * (1 + τ -1 + β -1 ) + 1)/(2 + d * /β), ε n ∝ n -β/(2β+d * ) (log n) t , εn ∝ n -β/(2β+d * ) (log n) t0 .
Proof. For all n 1, let α n = α 0 n 2/(2β+d * ) (log n) -2t/β with α 0 > 0 a sufficiently large constant. By [START_REF] Shen | Adaptive Bayesian multivariate density estimation with Dirichlet mixtures[END_REF]Theorem 4],

µ π : KL(π , π) α -β n , V 2 (π , π) α -β n e -cα 1/2 n (log αn) (2+β -1 )t
, where for all p > 0,

KL(π , π) = π (u) log π (u) π(u) du and V p (π , π) = π (u) log π (u) π(u) p du . (11)
Then, by Pinsker's inequality, π -π tv √ 2 KL(π , π) 1/2 and therefore

µ S ( √ 2α -β/2 n ) e -cα 1/2 n (log αn) (2+β -1 )t e -cnε 2 n ,
where S is defined by [START_REF] Douc | Asymptotics of the maximum likelihood estimator for general hidden Markov models[END_REF]. This concludes the proof using [START_REF] Shen | Adaptive Bayesian multivariate density estimation with Dirichlet mixtures[END_REF]Theorem 5].

Dirichlet process mixtures of Beta distributions Following [START_REF] Rousseau | Rates of convergence for the posterior distributions of mixtures of betas and adaptive nonparametric estimation of the density[END_REF], the posterior concentration rate established in Theorem 3 may be investigated in the case of mixtures of Beta distributions. For all α > 0 and all ε ∈ (0, 1), the probability density function of the Beta distribution with mean ε and scale parameter α is written:

g α,ε : x → x aα,ε-1 (1 -x) bα,ε-1 B(a α,ε , b α,ε ) , where a α,ε = α/(1 -ε), b α,ε = α/ε
[27] introduces the following prior distribution on density functions π = g α,P on (0, 1) with a random mixture distribution P :

(P, α) ∼ DP (ν) ⊗ π α , (13)
where π = g α,P and P = k j=1 p j δ εj . As in [START_REF] Rousseau | Rates of convergence for the posterior distributions of mixtures of betas and adaptive nonparametric estimation of the density[END_REF], assume that: -η is a probability distribution on (0, 1) such that there exist η , η > 0 and T 1 satisfying, for all x ∈ (0, 1), 0

< ηx T (1 -x) T η(x) η ;
-the density π α has support [n t , +∞) for some 0 < t < 1 and for all b 1 > 0, there exist c, c 1 , c 2 , c 3 > 0 such that for all α n satisfying α n n -t → +∞,

π α (c 1 α n < α < c 2 α n ) ce -b1α 1/2 n and π α (c 3 α n < α) ce -b1α 1/2 n .
Let β, L > 0 and H(β, L) be the class of Hölder functions with regularity β:

H(β, L) = f : (0, 1) → R : ∀(x, y) ∈ (0, 1) 2 , f ( β ) (x) -f ( β ) (y) L |x -y| β-β .
Consider also the following assumption.

H5 -(β) For all x ∈ (0, 1) π (x) > 0 and there exist β, L > 0 and 0

k 0 , k 1 < β such that π ∈ H(β, L) , π (k0) (0) > 0 , π (k1) (1) < 0 .
Corollary 5. Assume that H5-(β) holds with 0 < β t -1/2. Then, there exist constants c 1 , c 2 such that the prior distribution (13) satisfies H2-(c 1 , c 2 ) with, for some sufficienty large constant c 0 and all n 1,

ε n = c 0 n -β/(2β+1) (log n) 5β/(2β+1) , εn = n -β/(2β+1) (log n) 5β/(4β+2) .
Proof. In [START_REF] Rousseau | Rates of convergence for the posterior distributions of mixtures of betas and adaptive nonparametric estimation of the density[END_REF], the author analyzed the approximation of the probability density π on (0, 1) by a continuous mixture of the form [START_REF] Dubarry | Fast computation of smoothed additive functionals in general state-space models[END_REF]. When π is Hölder with regularity β > 0, it is shown in [27, Theorem 3.1] that for all α > 0 and all p > 1, there exists a probability density π α on (0, 1) such that KL(π , g α,πα ) cα -β , V p (π , g α,πα ) cα -β and π -g α,πα ∞ α -β/2 where KL and V p are defined in [START_REF] Douc | Asymptotic properties of the maximum likelihood estimator in autoregressive models with markov regime[END_REF]. If 0 < β 2 we may choose π α = π and, if β > 2, π α must be different from π to obtain the optimal approximation. Therefore, if π is Hölder with regularity β > 0, π can be sharply approximated by a continuous mixture of Beta distributions of the form [START_REF] Dubarry | Fast computation of smoothed additive functionals in general state-space models[END_REF]. Then, [START_REF] Rousseau | Rates of convergence for the posterior distributions of mixtures of betas and adaptive nonparametric estimation of the density[END_REF] introduced a discrete mixture approximation of g α,πα of the form g α,Pα where P α is a probability distribution with finite support. Using this distribution, it is shown in [27, page 171] that

µ π : KL(π , π) α -β n , V 2 (π , π) α -β n e -cα 1/2 n (log αn) 5/2
, for some positive constant c. Then, by Pinsker inequality, π -

π tv √ 2 KL(π , π) 1/2 and µ S ( √ 2α -β/2 n ) e -cN0α 1/2 n (log αn) 5/2 e -cnε 2 n ,
where εn = α

-β/2 n = ε 0 n -β/(2β+1) (log n) 5β/(4β+2)
and where S is defined by [START_REF] Douc | Asymptotics of the maximum likelihood estimator for general hidden Markov models[END_REF]. On the other hand, [27, proof of theorem 2.2, pages 171 to 173] introduces a sequence of subsets

(Π n ) n 1 such that, µ (Π c n ) Cα β n e -α √ αn log(αn) 5/2 e -cnε 2 n
and with a log-entropy of Π n in total variation distance lower than nε 2 n .

4 Numerical experiments

Block Gibbs sampling algorithm

Following [START_REF] Yau | Bayesian non-parametric hidden Markov models with applications in genomics[END_REF], a Markov Chain Monte Carlo (MCMC) method can be introduced to solve the Bayesian nonparametric problem introduced in this paper, i.e. to sample from the posterior distribution given X 1:n . The unknown distribution π is specified as a mixture model in which some probability density ϕ z is mixed with respect to a discrete probability measure P . Defining the discrete probability measure P as in [START_REF] Yau | Bayesian non-parametric hidden Markov models with applications in genomics[END_REF], the mixture of Dirichlet processes with base measure αQ, where α is a positive constant and Q a probability distribution, is given by:

(ϑ j ) j 1 ∼ i.i.d. Beta(1, α) , (14
)

ω 1 = ϑ 1 and for j 2 , ω j = ϑ j j-1 i=1 (1 -ϑ i ) , (15)
(z j ) j 1 ∼ i.i.d. Q , (16)
(κ i , u i ) 1 i n+1 ∼ i.i.d. j 1 1 ui<ωj δ j (κ i ) , (17)
V i ∼ ϕ zκ i for 1 i n + 1 , (18)
The Dirichlet process prior is defined by the stick-breaking weights (ϑ j ) j 1 , the mixture parameters (z j ) j 1 the allocation variables (κ i ) 1 i n+1 ans some auxiliary variables (u i ) 1 i n+1 . This representation with auxiliary variables is introduced in [START_REF] Walker | Sampling the Dirichlet mixture model with slices[END_REF] and used in [START_REF] Yau | Bayesian non-parametric hidden Markov models with applications in genomics[END_REF]. For all 1 i n + 1, integrating out the random variable u i on (0, 1) leads to the usual marginal distribution for κ i :

κ i ∼ i.i.d. j 1 ω j δ j ,
and moreover integrating out the random variable κ i , marginally

V i ∼ j 1 ω j ϕ zj = ϕ z (•)P (dz) ,
where P = j 1 ω j δ j is distributed from a Dirichlet process with base measure αQ. The auxiliary variables greatly simplifies the expression of the posterior distributions of κ i as detailed in (ii) below. Sampling from the joint posterior distribution of (V 1:n+1 , u 1:n+1 , κ 1:n+1 , z, ϑ) is then performed by block Gibbs sampling according to the following steps, where z = (z j ) j 1 and ϑ = (ϑ j ) j 1 .

(i) The hidden states V 1:n+1 are sampled according to their posterior distribution given the random variables (X 1:n , u 1:n+1 , z, ϑ) with a collapsed step. Note that by integrating out κ 1:n+1 , the conditional distribution of (X 1:n , V 1:n+1 ) given (u 1:n+1 , z, ϑ) is given by

p n (V 1:n+1 , X 1:n |u 1:n+1 , z, ϑ) ∝ n+1 i=1   j, ui<ωj ϕ zj (V i )   n i=1 K(X i , V i , V i+1 ) .
Therefore, the posterior distribution of V 1:n+1 given the random variables (X 1:n , u 1:n+1 , z, ϑ) is the joint smoothing distribution of V 1:n+1 given X 1:n when (V i ) 1 i n+1 are independent with V i ∼ j, ui<ωj ϕ zj for all 1 i n + 1. This step cannot be done explicitly as in [START_REF] Yau | Bayesian non-parametric hidden Markov models with applications in genomics[END_REF] where the hidden states are discrete. A Sequential Monte Carlo smoother described below is used instead.

(ii) For all 1 i n + 1, the posterior distribution of κ i given (X

1:n , V 1:n+1 , u 1:n+1 , ϑ, z) is given by κ i |X 1:n , V 1:n+1 , u 1:n+1 , ϑ, z ∼ j, ωj >ui ϕ zj (V i )δ j (κ i ) .
This expression motivates the use of the auxiliary variables u 1:n+1 . In the case where u 1:n+1 are not introduced in the model, κ i is sampled from its posterior distribution given (X 1:n , κ 1:n+1 , z, ϑ) which is proportional to j 1 ω j ϕ zj (V i )δ j (κ i ). Due to the infinite sum, this distribution has an intractable normalizing constant so that sampling κ i without the auxiliary variable u i is challenging.

(iii) (ϑ, u 1:n+1 ) are sampled according to their posterior distribution given (κ 1:n+1 , X 1:n , V 1:n+1 , z).

As detailed in [START_REF] Yau | Bayesian non-parametric hidden Markov models with applications in genomics[END_REF], (ϑ j ) j 1 are updated according to their distribution given (κ 1:n+1 , X 1:n , V 1:n+1 , z):

ϑ j |κ 1:n+1 , X 1:n , V 1:n+1 , z ∼ ind. Be m j + 1, n + 1 - j =1 m + α ,
where m j = Card({1 i n + 1 ; k i = j}). Then, the random variables (u i ) 1 i n+1 are conditionally independent given (κ 1:n+1 , α, X 1:n , V 1:n+1 , z, ϑ) and such that u i ∼ U (0, ω ki ).

(iv) z are sampled according to their posterior distribution given (X

1:n , κ 1:n , V 1:n+1 , u 1:n+1 , ϑ).
This posterior distribution is given, for all j 1 by

z j |X 1:n , κ 1:n , V 1:n+1 , u 1:n+1 , ϑ ∼ ind. Q(z j ) n+1 i=1, κi=j ϕ zj (V i ) .
Following [38, Section 4.2.2], the probability density function ϕ z is set as a Gaussian probability density with mean µ and variance 1/λ where z = (µ, λ). The distribution Q is chosen as Q = N (0, 1)⊗Gamma(1, 1), a standard Gaussian distribution for µ which is independent of the precision parameter λ distributed according to a Gamma(1, 1).

Sequential Monte Carlo smoother for step (i)

A Sequential Monte Carlo smoother is used to sample the hidden states V 1:n+1 according to their posterior distribution given the random variables (X 1:n , u 1:n , z, ϑ). This is the joint smoothing distribution of V 1:n+1 given X 1:n when marginally (V i ) 1 i n+1 are independent with V i ∼ j, ui<ωj ϕ zj for all 1 i n + 1.

Particle filtering Particle filtering algorithms are simulation based procedures used to approximate recursively the distributions

φ k of V k given X 1:k-1 for k = 1 to k = n + 1: φ k [h] = E [h(V k )|X 1:k-1 ] .
The auxiliary particle filter of [START_REF] Liu | Sequential Monte Carlo methods for dynamic systems[END_REF][START_REF] Pitt | Filtering via simulation: Auxiliary particle filters[END_REF] is the most widely used particle filtering method and is a generalization of other approaches such as the algorithms proposed in [START_REF] Gordon | Novel approach to nonlinear/non-Gaussian bayesian state estimation[END_REF] and [START_REF] Kitagawa | Monte-Carlo filter and smoother for non-Gaussian nonlinear state space models[END_REF]. The auxiliary particle filter produces recursively a set of states (ξ k ) M =1 associated with importance weights (ω k ) M =1 using importance sampling and resampling steps. At k = 1, (ξ 1 ) M =1 are sampled independently from ϕ zκ 1 and each particle ξ 1 is associated with the standard importance sampling weight ω 1 = 1/M . For any bounded and measurable function h defined on X, the expectation φ 1 [h] is approximated by a self normalized importance sampling estimator:

φ M 1 [h] = M -1 M =1 h ξ 1 .
Then, using {(ξ k-1 , ω k-1 )} M =1 , the auxiliary particle filter of [START_REF] Liu | Sequential Monte Carlo methods for dynamic systems[END_REF][START_REF] Pitt | Filtering via simulation: Auxiliary particle filters[END_REF] samples pairs {(I k , ξ k )} M =1 of superscripts and particles from the instrumental distribution:

π k ( , x) ∝ ω k-1 ρ k (ξ k-1 )p k (ξ k-1 , x) defined on {1, . . . , M } × V, where (ρ k (ξ k-1 )) M
=1 are adjustment multiplier weights and p k is a transition density chosen by the user. In the definition of π k , the weights ω k-1 ρ k (ξ k-1 ), 1 M , are used to duplicate promising particles or discard particles with low importance weights at the previous time step. For ∈ {1, . . . , M }, ξ k is associated with the following importance weight:

ω k = j, u k <ωj ϕ zj (ξ k ) K(X k , ξ I k k-1 , ξ k ) ρ k (ξ k-1 )p k (ξ I k k-1 , ξ k ) . ( 19
)
Particle smoothing The Forward Filtering Backward Simulation (FFBSi) algorithm proposed by [START_REF] Godsill | Monte Carlo smoothing for non-linear time series[END_REF] (see also [START_REF] Del Moral | A Backward Particle Interpretation of Feynman-Kac Formulae[END_REF][START_REF] Douc | Sequential Monte Carlo smoothing for general state space hidden Markov models[END_REF][START_REF] Dubarry | Nonasymptotic deviation inequalities for smoothed additive functionals in nonlinear state-space models with applications to parameter estimation[END_REF] for its convergence properties and [START_REF] Dubarry | Fast computation of smoothed additive functionals in general state-space models[END_REF] for its computational efficiency) samples backward trajectories from time n + 1 to time 1 among all the M n+2 trajectories which can be built using the particles ξ k , with 1 M and 1 k n + 1. Each trajectory is sampled according to the following steps:

-At time k = n + 1, sample J n+1 ∈ {1 . . . , M } with probabilities proportional to ω n+1 , for ∈ {1, . . . , M }.

-For k = n to k = 0, define for all 1 M ,

Λ M k (J k+1 , ) = ω k K(X k , ξ k , ξ J k +1 k+1 ) M j=1 ω j k K(X k , ξ j k , ξ J k +1 k+1 ) , 1 M . (20)
Then, J k is sampled in {1 . . . , M } with probabilities Λ M k (J k+1 , ), 1 M .

These sampling steps produced a trajectory (ξ J1 0 , . . . , ξ Jn+1 n+1 ) approximately distributed according to the target joint smoothing distribution. The computational complexity of the FFBSi algorithm grows with M 2 due to the normalizing constant of Λ M k , 1 k n. In the context of this paper, K is upper bounded by 1 and the acceptance rejection mechanism introduced in [START_REF] Douc | Sequential Monte Carlo smoothing for general state space hidden Markov models[END_REF] to implement the FFBSi algorithm with a complexity which grows linearly with M can be used. In this case, at each time step 1 k n, the new index J k is sampled in {1, . . . , M } with probabilities proportional to (ω k ) M =1 and accepted with probability

K(X k , ξ J k k , ξ J k+1 k+1 ).

Application to soccer results

The algorithm introduced in Section 4.1 is used in this section to estimate the distribution of French soccer teams abilities. Using the outcomes (wins, losses, ties) of the first 30 games during the season 2017-2018, the Expectation Maximization (EM) algorithm for generalized Bradley Terry model with home advantage and ties described in [START_REF] Caron | Efficient Bayesian inference for generalized Bradley-Terry models[END_REF] is run to estimate the abilities of the 20 teams with X = {-1, 0, 1} and the distribution K(1, v i , v j ) = αe vi αe vi + θe vj and K(0, v i , v j ) = (θ 2 -1)αe vi e vj (αe vi + θe vj )(θαe vi + e vj ) .

The parameters α, characterizing the home advantage (v i is the home team), and θ, associated with ties, are also estimated with the EM algorithm. The estimates (v i ) 1 i 20 , α and θ are set as the estimates produced by the algorithm btemhometies1 , see Figure 3. The (supposedly) unknown target density π for the Block Gibbs sampling algorithm is then defined as kernel density estimate of the strengths (rescaled around zero) (v i ) 1 i 20 obtained with the function ksdensity of Matlab, see Figure 3.

Then n = 3000 observations are sampled by first sampling independently (V i ) 1 i n+1 distributed according to this target density and the observation model [START_REF] Kitagawa | Monte-Carlo filter and smoother for non-Gaussian nonlinear state space models[END_REF] with α and θ. The initial estimate of the Block Gibbs sampling algorithm is set as a mixture of two Gaussian distributions N (µ 1 , σ 2 1 ) and N (µ 2 , σ 2

2 ) with µ 1 and µ 2 independent with distribution N (0, 1) and σ -2 1 and σ -2 2 independent with distribution Gamma(1, 1). The weight of the first Gaussian distribution is distributed as a Beta(1, α) random variable with α = 1.

The Sequential Monte Carlo smoother is set as a standard Bootstrap filter i.e. at each time step 1 k n + 1, ρ k = 1 and p k (x, x ) = j ,,u k <ωj ϕ zj (x ). The number of particles is set to M = 100 and 6500 iterations of the Block Gibbs sampling algorithm are performed. The first 5000 samples are discarded as a burn-in period and the estimate of π is set as the average over the last 1500 samples produced by the Block Gibbs Sampler, see Figure 4. By [START_REF] Kitagawa | Monte-Carlo filter and smoother for non-Gaussian nonlinear state space models[END_REF], the law of the observations only depends on (e Vi ) 1 i n+1 , therefore π can only be identified up to a translation on R. Such a translation is displayed in Figure 4 to highlight that the Block Gibbs Sampler recovers mainly the shape of π .

In Figure 5 and Figure 6 1000 championships are sampled independently using for each run 20 parameters sampled with the estimated density obtained with the Block Gibbs Sampler to produce boxplots of the scores at the end of the championship. These results are compared to the scores of 1000 championships run with the parameter estimates provided by btemhometies in Figure 5 and with parameters sampled from the target density in Figure 6.

A Proof of Theorem 3 A.1 Proof of Proposition 1

Note first that

E π [log P π (X 1:n ) -log P π (X 1:n )] = n i=1 E π log P π (X i |X 1:i-1 ) P π (X i |X 1:i-1 ) , = n i=1 E π xi∈X P π (X i = x i |X 1:i-1 ) log P π (X i = x i |X 1:i-1 ) P π (X i = x i |X 1:i-1 )
.

The inner term of the expectation may be upper bounded using that for two probability distributions p and q on X , Then, by Lemma 8,

x∈X p(x) log p(x) q(x) x∈X p(x) p(x) -q(x) q(x) = x∈X (p(x) -q(x)) 2 q(x) .
xi∈X P π (X i = x i |X 1:i-1 ) log P π (X i = x i |X 1:i-1 ) P π (X i = x i |X 1:i-1 ) xi (P π (X i = x i |X 1:i-1 ) -P π (X i = x i |X 1:i-1 )) 2 P π (X i = x i |X 1:i-1 ) , 4|X |ν -5 (2 + ν -1 ) 2 π -π 2
tv , which concludes the proof of [START_REF] Chetrite | The number of potential winners in bradley-terry models in random environment[END_REF]. By [START_REF] Vernet | Non parametric hidden markov models with finite state space: Posterior concentration rates[END_REF]Equation (32)],

Var π [log P π (X 1:n ) -log P π (X 1:n )] 4 1 i n E π Z 2 i + 4 1 i<j n E π Z 2 j 1/2 E π (E π [Z j |X 1:i-1 ] -E π [Z j ]) 2 1/2 , ( 22
)
where, for all i 1, Z i = log P π (X i |X 1:i-1 ) -log P π (X i |X 1:i-1 ). Following [START_REF] Douc | Asymptotics of the maximum likelihood estimator for general hidden Markov models[END_REF], ( 22) can be upper bounded using the extended Markov chain defined, for all 1 i n, by

R i = (V i+1 , X i , η π i , η π i ) , (23)
where, for all i 1, η π i is the predictive filter at time i when the law of the hidden states is π:

η π 1 = π and for i 2, η π i : A → P π (V i ∈ A|X 1:i-1
) . Define the function h, for all r = (v, x, η, η ) ∈ X × V × S + × S + , by:

h(r) = log π (dv p+1 )η (dv p )K(v p , v p+1 , x) π(dv p+1 )η(dv p )K(v p , v p+1 , x) . (24)
Then, for all i ≥ 1,

h(R i ) = log P π (X i |X 1:i-1 ) -log P π (X i |X 1:i-1 ) = Z i
and for all 1 i < j n,

|E π [Z j |X 1:i-1 ] -E π [Z j ]| = |E π [h(R j )|X 1:i-1 ] -E π [h(R j )]| .
Let Φ be defined for any π ∈ Π, x ∈ X and probability distribution η on V by:

Φ(x, η; π) : A → 1 A (z)π(dz)η(dz )K(z , z, x) π(dz)η(dz )K(z , z, x) .
For all i 2, the predictive filter at time i may be expressed as:

η π i = Φ(X i-1 , η π i-1 ; π) . ( 25
)
The transition kernel of the extended chain is given, for all (x, v, η, η ) ∈ V × X × S + × S + and all B(V) × P(X ) × B(S + ) × B(S + ), by

Q π (v, x, η, η , A V , A X , B 1 , B 2 ) = x ∈X π(dv )K(v, v , x )1 A X (x )1 A V (v )1 B1 (Φ(x, η; π))1 B2 (Φ(x, η ; π )) . ( 26
)
For all p 1, the p -th iterate of Q π is denoted by Q p π . For all 1 i < j n, using Lemma 10,

|E π [Z j |X 1:i-1 ] -E π [Z j ]| Q j-i π h(r) -Q j-i π h(r) p π (dr i |X 1:i-1 ) p π (dr i ) , 6ν -1 (1 -ν) j-i-2 . ( 27
)
On the other hand, by Lemma 8, for all 1 j n,

E π Z 2 j 4ν -6 (2 + ν -1 ) 2 π -π 2 tv , (28)
so that for all 0 < β < 1, using ( 22), ( 27) and ( 28),

Var π [log P π (X 1:n ) -log P π (X 1:n )] c ν 1 i n E π Z 2 i + c ν 1 i<j n E π Z 2 j |E π [Z j |X 1:i-1 ] -E π [Z j ]| β E π Z 2 j 1-β/2 , c ν β -2 π -π 2-β/2 tv n ,
where c ν is a constant which depends on ν only. The proof is completed by noting that the best upper bound is obtained for β = -4/ log( π -π tv ).

A.2 Proof of Proposition 2

For all π, π , π ∈ Π, by [START_REF] Douc | Sequential Monte Carlo smoothing for general state space hidden Markov models[END_REF],

∆ n π,π (X 1:n ) -∆ n π,π (X 1:n ) = 1 n n (π , X 1:n ) - 1 n n (π , X 1:n ) -L π (π ) + L π (π ) , = 1 n n (π , X 1:n ) -E π 1 n n (π , X 1:n ) + E π 1 n n (π , X 1:n ) -L π (π ) - 1 n n (π , X 1:n ) + E π 1 n n (π , X 1:n ) -E π 1 n n (π , X 1:n ) + L π (π ) .
Note first that by Lemma 7, for all π ∈ Π,

sup π ∈ Π E π 1 n n (π , X 1:n ) -L π (π ) 1 nν 2 . Therefore, for all π ∈ Π, sup π ,π ∈ Π ∆ n π,π (X 1:n ) -∆ n π,π (X 1:n ) 2 nν 2 + 2G n π (X 1:n ) , (29)
where

G n π (X 1:n ) = sup π ∈ Π Z n π,π (X 1:n ) , Z n π,π (X 1:n ) = n -1 n (π , X 1:n ) -n -1 E π [ n (π , X 1:n )] .
G n π is a function of the Markov chain (X i , V i+1 ) 1 i n whose transition kernel P π is uniformly lower bounded by [START_REF] Cappe | Inference in hidden Markov models[END_REF]. By Corollary 12, it is enough to obtain a bounded difference inequality to establish a concentration inequality for G n π . For all 1 k n and all x 1:n ∈ X n define x(k) ∈ X n such that

x(k) k = x k and for all j = k , x(k) j = x j . (30)
Note that, for any π ∈ Π,

Z n π,π (x 1:n ) -Z n π,π (x (k) 1:n ) n -1 n (π , x 1:n ) -n (π , x(k) 1:n ) , n -1 n i=1 log P π (x i |x 1:i-1 ) -log P π (x (k) i |x (k) 1:i-1 ) , n -1 n i=k log P π (x i |x 1:i-1 ) -log P π (x (k) i |x (k) 1:i-1 ) , as for 1 i k -1, x i = x(k) i . In the case i = k, |P π (x i |x 1:i-1 ) -P π (x (k) i |x (k) 1:i-1 )| = |P π (x k |x 1:k-1 ) -P π (x (k) k |x 1:k-1 )| 1 , so that | log P π (x i |x 1:i-1 ) -log P π (x (k) i |x (k) 1:i-1 )| ν -1
. For all k + 1 i n, by Lemma 6,

P π (x (k) i |x (k) 1:i-1 ) = P π (dv k+1 |x (k) 1:i-1 ) i-1 j=k+1 K V |X π ,j,i-1 (v j , dv j+1 )π (dv i+1 )K(v i , v i+1 , x i ) and P π (x i |x 1:i-1 ) = P π (dv k+1 |x 1:i-1 ) i-1 j=k+1 K V |X π ,j,i-1 (v j , dv j+1 )π (dv i+1 )K(v i , v i+1 , x i ) .
Therefore, |P

:i-1 ) -P π (x i |x 1:i-1 )| (1 -ν) i-k-1 and | log P π (x π (x (k) i |x (k) 1
:i-1 ) -log P π (x i |x 1:i-1 )| ν -1 (1 -ν) i-k-1 (k) i |x (k) 1
which yields, for any π ∈ Π,

Z n π,π (x 1:n ) -Z n π,π (x (k) 1:n ) n -1 ν -1 + n i=k+1 ν -1 (1 -ν) i-k-1 2 nν 2 .
By Corollary 12 applied with γ 1 = . . . = γ n = 2/(nν 2 ), for all π, π ∈ Π and all t > 0,

P π Z n π,π (X 1:n ) 2 √ 5t ν 3 √ n 2e -t 2 . ( 32
)
By Lemma 8,

|Z π,π1 (x 1:n ) -Z π,π2 (x 1:n )| 4ν -3 2 + ν -1 π 1 -π 2 tv . (33)
Let R(ε) denote an ε-net of Π. By [START_REF] Vapnik | Statistical learning theory[END_REF],

G n π (X 1:n ) max π ∈R(ε) |Z π,π (x 1:n )| + 4ν -3 2 + ν -1 ε .
Applying a union bound to [START_REF] Alexandre | Optimal aggregation of classifiers in statistical learning[END_REF] yields

P π G n π (X 1:n ) 2 √ 5t ν 3 √ n + 4ν -3 2 + ν -1 ε 2N( Π, . tv , ε)e -t 2 .

A.3 Proof of Theorem 3

For n 1,

µ B c c, (ε n ) X 1:n = B c c, (εn) exp { n (π, X 1:n ) -n (π , X 1:n )} µ(dπ) Π exp { n (π, X 1:n ) -n (π , X 1:n )} µ(dπ) = N n D n .
Following [28, Section 3.3], consider, for all u > 0 and all function g : R + → R + , the decomposition

P π µ B c c, (ε n ) X 1:n > α n E π Φ τn,c(u) n (X 1:n , π ) + P π (D n < g(u)
)

+ P π Φ τn,c(u) n (X 1:n , π ) = 0 ∩ {D n ≥ g(u)} ∩ N n D n > α n ,
where for all n 1 and all t > 0,

Φ t n (X 1:n , π ) = 1 {∃π∈Πn : n (π,X1:n)/n-n (π ,X1:n)/n+Lπ (π )-Lπ (π)>t} (34)
and for all u > 0, c > 0 and n 1

τ n,c (u) = c n -1/2 u + ε n .
By Markov inequality, for all u > 0,

P π Φ τn,c(u) n (X 1:n , π ) = 0 ∩ {D n ≥ g(u)} ∩ N n D n > α n P π Φ τn,c(u) n (X 1:n , π ) = 0 ∩ {N n > α n g(u)} , P π B c c, (εn) 1 -Φ τn,c(u) n (X 1:n , π ) exp { n (π, X 1:n )} exp { n (π , X 1:n )} µ(dπ) > α n g(u) , 1 α n g(u) E π B c c, (εn) 1 -Φ τn,c(u) n (X 1:n , π ) exp { n (π, X 1:n )} exp { n (π , X 1:n )} µ(dπ) , 1 α n g(u) B c c, (εn) E n π 1 -Φ τn,c(u) n (X 1:n , π ) µ(dπ) .
Therefore,

P π µ B c c, (ε n ) X 1:n > α n E π Φ τn,c(u) n (X 1:n , π ) + P π (D n < g(u)) + 1 α n g(u) B c c, (εn)∩Πn E π 1 -Φ τn,c(u) n (X 1:n , π ) µ(dπ) + µ (Π c n ) α n g(u)
.

Then,

E π Φ τn,c(u) n (X 1:n , π ) = P π (∃π ∈ Π n : ( n (π, X 1:n ) -n (π , X 1:n ))/n -(L π (π) -L π (π )) > τ n,c (u)) , P π ∃π ∈ Π n : ∆ n π ,π (X 1:n ) -∆ n π ,π (X 1:n ) > τ n,c (u) ,
where ∆ n π ,π (X 1:n ) and ∆ n π ,π (X 1:n ) are defined by [START_REF] Douc | Sequential Monte Carlo smoothing for general state space hidden Markov models[END_REF]. Then, by Proposition 2 with Π = Π n ∪{π }, choosing c = (1 + √ 2)c ν and u = u n = (c/(2c) -1) √ nε n , for any sufficiently large n,

E π Φ τn,c(un) n (X 1:n , π ) e -u 2 n .
This yields

P π µ B c c, (ε n ) X 1:n > α n e -u 2 n + P π (D n < g(u n )) + 1 α n g(u n ) B c c, (εn)∩Πn E π 1 -Φ τn,c(un) n (X 1:n , π ) µ(dπ) + µ (Π c n ) α n g(u n ) .
Then, using that for all π ∈ Π n , L π (π)-L π (π ) 0, and for all π

∈ B c c, (ε n ), L π (π )-L π (π) > cε n , E π 1 -Φ τn,c(un) n (X 1:n , π ) = P π (∀π ∈ Π n : n (π , X 1:n )/n -n (π , X 1:n )/n -(L π (π ) -L π (π )) τ n,c (u n )) , P π ( n (π, X 1:n )/n -n (π , X 1:n )/n τ n,c (u n ) -cε n ) , P π (( n (π, X 1:n ) -n (π , X 1:n ))/n -(L π (π) -L π (π )) τ n,c (u n ) -cε n ) . Since τ n,c (u n ) ≤ cε n /2, E π 1 -Φ τn,c(un) n (X 1:n , π ) P π ∆ n π,π (X 1:n ) -∆ n π,π (X 1:n ) cε n /2 , P π ∆ n π,π (X 1:n ) -∆ n π,π (X 1:n ) τ n,c (u n ) , e -u 2 n ,
where the last inequality follows from Proposition 2. Then,

P π µ B c c, (ε n ) X 1:n > α n e -u 2 n + P π (D n < g(u n )) + e -u 2 n α n g(u n ) + µ (Π c n ) α n g(u n ) .
Let c 3 > c ν where c ν is given in Proposition 1. Since

D n S (εn) exp { n (π, X 1:n ) -n (π , X 1:n )} 1 { n(π,X1:n )-n (π ,X1:n) -c3nε 2 n } µ(dπ) , then P π (D n < g(u n )) P π µ S (ε n ) ∩ π : n (π, X 1:n ) -n (π , X 1:n ) -c 3 nε 2 n < g(u n )e c3nε 2 n , P π µ S (ε n ) ∩ π : n (π, X 1:n ) -n (π , X 1:n ) -c 3 nε 2 n c > µ (S (ε n )) -g(u n )e c3nε 2 n .
There exists a function π such that, for all distribution π , P π -a.s. and in L 1 (P π ),

1 n n (π, X 1:n ) -→ n→∞ L π (π) = E π [ π (X)]
and

E π 1 n n (π, X 1:n ) -L π (π) 1 nν 2 .
Proof. Let x ∈ X Z . For all i 2 and all 0,

P π (x i |x 1:i-1 ) = P π (dv 1 |x 1:i-1 ) i-1 k=1 K V |X π,k,i-1 (v k , dv k+1 ) π(dv i+1 )K(x i , v i , v i+1 ) , P π (x i |x -:i-1 ) = P π (dv 1 |x -:i-1 ) i-1 k=1 K V |X π,k,i-1 (v k , dv k+1 ) π(dv i+1 )K(x i , v i , v i+1 ) ,
as V 2:i+1 is independent of X -:0 conditionally on V 1 . Therefore, by Lemma 6,

|P π (x i |x 1:i-1 ) -P π (x i |x -:i-1 )| (1 -ν) i-1
and, since P π (x i |x 1:i-1 ) ∧ P π (x i |x -:i-1 ) ν,

|log P π (x i |x 1:i-1 ) -log P π (x i |x -:i-1 )| ν -1 (1 -ν) i-1 .
Similarly, for i = 1,

|log P π (x 1 ) -log P π (x 1 |x -:0 )| ν -1 .
Let (X i , V i+1 ) i∈Z be a process with distribution P π . For all i ∈ Z,

|log P π (X i |X 1:i-1 ) -log P π (X i |X -:i-1 )| ν -1 (1 -ν) i-1 .
Define the shift operator ϑ on X Z by (ϑx) i = x i+1 for all i ∈ Z and all x ∈ X Z . There exists a function π such that for all i ∈ Z the sequence (log P π (X i |X -:i-1 )) 0 converges P π -a.s. to π ϑ i X . Then, when grows to ∞, this yields

log P π (X i |X 1:i-1 ) -π ϑ i X ν -1 (1 -ν) i-1 .
Then,

1 n n (π, X 1:n ) - 1 n n i=1 π ϑ i X = 1 n n i=1 {log P π (X i |X 1:i-1 ) -π ϑ i X } 1 nν 2
and the proof of the second claim is completed with the ergodic theorem [START_REF] Billingsley | Probability and Measure[END_REF]Theorem 24.1]. Remarking that for all i 1, E π π ϑ i X = L π (π), the loglikelihood of X 1:n also satisfies

E π 1 n n (π, X 1:n ) -L π (π) = E π 1 n n i=1 log P π (x i |x 1:i-1 ) -π ϑ i X 1 nν 2 .
Lemma 8. Assume that H1 holds. For all x 1:n ∈ X n , all distributions π, π ∈ Π and any 1 i n,

|P π (x i |x 1:i-1 ) -P π (x i |x 1:i-1 )| 2ν -2 (2 + ν -1 ) π -π tv .
Proof. When i = 1,

P π (x 1 ) -P π (x 1 ) = {π ⊗ π(dv 1:2 ) -π ⊗π (dv 1:2 )} K(x 1 , v 1 , v 2 ) .
Thus |P π (x 1 ) -P π (x 1 )| ≤ 2 π -π tv . For all 2 i n,

P π (x i |x 1:i-1 ) -P π (x i |x 1:i-1 ) = i+1 =1 {P (x i |x 1:i-1 ) -P +1 (x i |x 1:i-1 )} ,
where P is the joint distribution of (X 1:i , V 1:i+1 ) when (V 1 , . . . , V -1 ) are i.i.d. with distribution π and (V , . . . , V i+1 ) are i.i.d. with distribution π. Then, for all 1 i -1,

P (x i |x 1:i-1 ) = P (dv +1 |x 1:i-1 ) i k= +1 K V |X π,k,i-1 (v k , dv k+1 ) K(x i , v i , v i+1 ) , = P (dv +1 |x 1:i-1 ) i-1 k= +1 K V |X π,k,i-1 (v k , dv k+1 ) π(dv i+1 )K(x i , v i , v i+1 ) .
Therefore, for all 1 i -1, by Lemma 6,

|P (x i |x i+1:n ) -P +1 (x i |x i+1:n )| ≤ (1 -ν) i--1 P (•|x 1:i-1 ) -P +1 (•|x 1:i-1 ) tv ,
where

P (•|x 1:i-1 ) is the distribution of V +1 conditionally on {X 1:i-1 = x 1:i-1 } when (V 1 , . . . , V -1 )
are i.i.d. with distribution π and (V , . . . , V i+1 ) are i.i.d. with distribution π. We first show that

P (•|x 1:i-1 ) -P +1 (•|x 1:i-1 ) tv ≤ 2ν -2 π -π tv .
Write, for all 1 i -1,

L x1:i-1 (dv 1:i ) = -1 m=1 π (dv m ) i m= π(dv m ) i-1 m=1 K(x m , v m , v m+1 ) . (35)
Then, for any function f on V,

f (v +1 )P (dv +1 |x 1:i-1 ) = f (v +1 )L x1:i-1 (dv 1:i ) L x1:i-1 (dv 1:i ) . Therefore, f (v +1 ) {P (dv +1 |x 1:i-1 ) -P +1 (dv +1 |x 1:i-1 )} = f (v +1 ) L x1:i-1 (dv 1:i ) L x1:i-1 (dv 1:i ) - L x1:i-1 +1 (dv 1:i ) L x1:i-1 +1 (dv 1:i ) , = f (v +1 ) L x1:i-1 (dv 1:i ) -L x1:q-1 +1 (dv 1:i ) L x1:i-1 (dv 1:i ) + f (v +1 ) L x1:i-1 +1 (dv 1:i ) L x1:i-1 +1 (dv 1:i ) L x1:i-1 +1 (dv 1:i ) -L x1:i-1 (dv 1:i ) L x1:i-1 (dv 1:i ) .
Thus, for any function

f on V such that f ∞ < 1, f (v +1 ) {P (dv +1 |x 1:i-1 ) -P +1 (dv +1 |x 1:i-1 )} ≤ 2 | {L x1:i-1 (dv 1:i ) -L x1:i-1 +1 (dv 1:i )}| L x1:i-1 (dv 1:i ) . (36)
By ( 35), 1 i -1,

{L x1:i-1 (dv 1:i ) -L x1:i-1 +1 (dv 1:i )} = -1 m=1 π (dv m ) {π(dv ) -π (dv )} i m= +1 π(dv m ) i-1 m=1 K(x m , v m , v m+1 ) .
As K is upper bounded by 1,

{L x1:i-1 (dv 1:i ) -L x1:i-1 +1 (dv 1:i )} ≤ -1 m=1 π (dv m ) -2 m=1 K(x m , v m , v m+1 ) × π -π tv i m= +1 π(dv m ) i-1 m= +1 K(x m , v m , v m+1 ) .
Similarly, since K is respectively lower bounded by ν,

L x1:i-1 (dv 1:i ) ≥ -1 m=1 π (dv m ) -2 m=1 K(x m , v m , v m+1 ) × ν 2 i m= +1 π(dv m ) i-1 m= +1 K(x m , v m , v m+1 ) .
Plugging these bounds in (36) yields, for 1 i -1,

f (v +1 ) {P (dv +1 |x 1:i-1 ) -P +1 (dv +1 |x 1:i-1 )} ≤ 2ν -2 π -π tv .
Then, by bounding similarly the two last terms of the telescoping sum,

P π (x i |x 1:i-1 ) -P π (x i |x 1:i-1 ) = 2ν -2 π -π tv i-1 =1 (1 -ν) i--1 + 2 ≤ 2(2 + ν -1 )ν -2 π -π tv .

C Forgetting properties of the predictive filter

For all k 1, η π k is the predictive filter at time k when the law of the hidden states is π: η

π 1 = π and for k ≥ 2, η π k : A → P π (V k ∈ A|X 1:k-1 )
.

By applying [START_REF] Mammen | Smooth discrimination analysis[END_REF] recursively, for all p 1, define the function Φ p by:

η π p+1 = Φ(X p , η π p ; π) = Φ(X p , Φ(X p-1 , η π p-1 ; π); π) = . . . = Φ p (X 1:p , η π 1 ; π) ,
with the convention Φ 1 = Φ. Lemma 9 establishes the exponential forgetting of the prediction filter. Its proof follows closely [9, Proposition 1] in the case of general hidden Markov models.

Lemma 9. Assume that H1 holds. For all p 1, all x 1:p ∈ X p and all probability distributions η, η , π on V, Φ p (x 1:p , η; π) -Φ p (x 1:p , η ; π) tv (1 -ν) p .

Proof. By definition, Φ p (x 1:p , η; π) is the predictive distribution of V p+1 given {X

1:p = x 1:p } when V 1 ∼ η and V k ∼ π for 2 ≤ k ≤ p + 1. By Lemma 6, Φ p (x 1:p , η; π)(dv p+1 )h(v p+1 ) = φ η,π,p (dv 1 ) p k=1 K V |X π,k,p (v k , dv k+1 )h(v p+1 ) , where φ η,π,p is the conditional distribution of V 1 given {X 1:p = x 1:p } when V 1 ∼ η and V k ∼ π for 2 ≤ k ≤ p + 1. Therefore, by Lemma 6, Φ p (x 1:p , η; π) -Φ p (x 1:p , η ; π) tv (1 -ν) p φ η,π,p -φ η ,π,p tv ,
which concludes the proof.

Lemma 10. Assume H1 holds. Then, for all r = (v, x, η, η

) ∈ V ×X ×S + ×S + , r = (v , x , η , η ) ∈ V × X × S + × S + and all p 1, |Q p π h(r) -Q p π h(r )| 6ν -1 (1 -ν) p-2 ,
where h and Q π are defined in [START_REF] Liu | Sequential Monte Carlo methods for dynamic systems[END_REF] and (26).

Proof. For all measurable function h, r = (v, x, η, η ) ∈ X × V × S + × S + , r = (v , x , η , η ) ∈ X × V × S + × S + and all p ≥ 1,

|Q p π h(r) -Q p π h(r )| ≤ |Q p π h(v, x, η, η ) -Q p π h(v, x, η , η )| + |Q p π h(v, x, η , η ) -Q p π h(v , x , η , η )| . (37)
With the convention v 2 = v, for all p 1,

Q p π h(r) = x2:p+1∈X p h(v p+2 , x p+1 , Φ p ((x, x 2:p ), η; π), Φ p ((x, x 2:p ), η ; π )) × p+2 k=3 K(x k-1 , v k-1 , v k )π ⊗p (dv 3:p+2 ) .
By H1, the function h defined in [START_REF] Liu | Sequential Monte Carlo methods for dynamic systems[END_REF] satisfies:

|h(v, x, η, η ) -h(v, x, η , η )| ν -1 ( η -η tv + η -η tv ) . (38)
The exponential forgetting property of the predictive filter given by Lemma 9 yields

Φ p-1 (x 2:p , Φ(x, η; π); π) -Φ p-1 (x 2:p , Φ(x, η ; π); π) tv (1 -ν) p-1 (39)
and, as Φ p ((x, x 2:p ), η; π) = Φ p-1 (x 2:p , Φ(x, η; π); π),

|Q p π h(v, x, η, η ) -Q p π h(v, x, η , η )| 2 ν (1 -ν) p-1 .
For the second term of (37), write for p 2, Φ p ((x, x 2:p ), η; π) = Φ p-2 (x 3:p , Φ 2 ((x, x 2 ), η; π); π)

and for any φ and φ in P,

h(v p+2 , x p+1 , Φ p ((x, x 2:p ), η; π), Φ p ((x, x 2:p ), η ; π )) = h(v p+2 , x p+1 , Φ p-2 (x 3:p , Φ 2 ((x, x 2 ), η; π); π), Φ p-2 (x 3:p , Φ 2 ((x, x 2 ), η ; π ); π ))
-h(v p+2 , x p+1 , Φ p-2 (x 3:p , φ; π), Φ p-2 (x 3:p , φ ; π ))

+ h(v p+2 , x p+1 , Φ p-2 (x 3:p , φ; π), Φ p-2 (x 3:p , φ ; π )) .

Then using that In the case p = 1, |Q p π h(v, x, η , η ) -Q p π h(v , x , η , η )| 4ν -1 .

D Concentration inequality for Markov chains with uniformly lower bounded transition kernel

Theorem 11 and its proof can be found in [START_REF] Douc | Markov Chains[END_REF]. As this book is not published yet, the proof of this result is reproduced here for the sake of completeness. Let ν > 0 and (X, B(X)) be a measurable space. In this section, P denotes a Markov kernel defined on (X, B(X)) bounded from below by ν > 0 : there exists a probability measure η on (X, B(X)) such that for all A ∈ B(X) and all x ∈ X, P (x, A) νη(A) .

(40)

For any x ∈ X (resp. any probability distribution π defined on (X, B(X))), let P x (resp. P π ) be the distribution of a Markov chain (X n ) n 0 with Markov kernel P and such that X 0 = x (resp. X 0 ∼ π). The expectation w.r.t. P x (resp. P π ) is denoted by E x (resp. E π ). Let γ ∈ R n + and B(γ) be the set of functions f : X → R such that, for all x, y ∈ X n , k ∈ {1, . . . , n},

|f (x) -f (y)| n k=1 γ k 1 x k =y k .
Following for instance [START_REF] Cappe | Inference in hidden Markov models[END_REF]Chapter 4], the Dobrushin coefficient of P can be defined as ∆(P ) = sup (x,x )∈X 2 P (x, •) -P (x , •) tv = sup π =π πP -π P tv π -π tv ,

where for any distribution π on (X, B(X)) and all A ∈ B(X), πP (A) = π(dx)P (x, A). For all n 1, define

D n = n =1 γ + 2 n m= +1 γ m ∆ P m- 2 . (41)
Theorem 11. For any distribution π on (X, B(X)), any γ ∈ R n + , any f ∈ B(γ) and any t > 0, P π (f (X 1 , . . . , X n ) -E π [f (X 1 , . . . , X n )] > t) e -2t 2 /Dn , where D n is defined by (41).

Proof. Assume without loss of generality that E π [f (X 1 , . . . , X n )] = 0. For any ∈ {1, . . . , n}, define

A = γ + 2 n m= +1
γ m ∆(P m-) .

For all 0, define F = σ(X i , 1 i ) and G = E π [f (X 1 , . . . , X n )|F ] with the conventions F 0 = {∅, Ω) and G 0 = 0. Hence,

f (X 1 , . . . , X n ) = n =1 (G -G -1 ), G -1 = E[G |F -1 ] .
The pivotal step of the proof is to establish that for all 1 n there exist F -1 -measurable random variables a and b such that

For all 1 n, s n -1, x 1: ∈ X and x s+1:n ∈ X n-s define K ,s (x 1 , . . . , x , x s+1 , . . . , x n ) = inf (u +1 ,...,us)∈X s- f (x 1 , . . . , x , u +1 , . . . , u s , x s+1 , . . . , x n ) and W ,s (x 1 , . . . , x , x s+1 , . . . , x n ) = K ,s (x 1 , . . . , x , x s+1 , . . . , x n ) -K ,s+1 (x 1 , . . . , x , x s+2 , . . . , x n ) .

Hence, 0 W ,s γ s+1 and G = K ,n (X 1 , . . . , X ) + n-1 s= E π [W ,s (X 1 , . . . , X , X s+1 , . . . , X n )|F ] .

Let X * 1 , . . . , X * n denote an independent copy of X 1 , . . . , X n so that Write for all y ∈ X, h(y) = E π [W ,s (X 1 , . . . , X , y)|F ] and h (y) = E y [h(X 1 , . . . , X n-s )] .

G = K ,n (

Then,

|E π [W ,s (X 1 , . . . , X , X s+1 , . . . , X n )|F ] -E π [W ,s (X 1 , . . . , X , X * s+1 , . . . , X * n )|F ]| = |E X [h(X s+1-, . . . , X n-)] -E πP [h(X s+1-, . . . , X n-)]| , = πP (dy)(P s+1-h (X ) -P s+1-h (y)) , h ∞ ∆(P s+1-) .

As h ∞ W ,s ∞ γ s+1 , we have

|G -H | n s= +1
γ s ∆(P s-) .

Define, for all 1 n, the following F -1 -measurable random variables a = inf γ s ∆(P s-)

Figure 1 :

 1 Figure 1: Graphical model of the nonparametric hidden graph.

Figure 2 :

 2 Figure 2: Graphical model of the nonparametric hidden graph.

Corollary 4 .

 4 x) ≤ b 4 e -b5|x| τ . Let π ∈ C(β, L, τ ). Assume that H3-(κ) and H4-(β, L) hold with κ > 0 and let d * = max(1, κ). Then, there exist c 1 , c 2 such that the prior distribution µ defined in (10) satisfies H2-(c 1 , c 2 ) for every positive constants c 1 and c 2 with, for all n 1,

  and for all a, b > 0, B(a, b) = Γ(a)Γ(b)/Γ(a + b), with Γ the Gamma function. For any probability distribution P on (0, 1) and any α > 0, define the continuous mixture g α,P by g α,P = g α,ε P (dε) .

Figure 3 :

 3 Figure 3: Results of the EM algorithm btemhometies with 2017-2018 Ligue 1 results.

Figure 4 :

 4 Figure 4: Density estimates from the Block Gibbs sampler.

Figure 5 :

 5 Figure 5: Estimated scores at the end of the championship with btemhometies parameter estimates: median (dotted line) and first and last deciles (grey area). Boxplots of the scores obtained with the Block Gibbs Sampler.

Figure 6 :

 6 Figure 6: Estimated scores at the end of the championship with parameter sampled with the target distribution: median (dotted line) and first and last deciles (grey area). Boxplots of the scores obtained with the Block Gibbs Sampler.

x2:p+1∈X p

  h(v p+2 , x p+1 , Φ p-2 (x 3:p , φ; π), Φ p-2 (x 3:p , φ ; π )) p+2 k=4 K(x k-1 , v k-1 , v k )K(x 2 , v, v 3 )π ⊗p (dv 3:p+2 ) = x2:p+1∈X p h(v p+2 , x p+1 , Φ p-2 (x 3:p , φ; π), Φ p-2 (x 3:p , φ ; π )) p+2 k=4 K(x k-1 , v k-1 , v k )K(x 2 , v , v 3 )π ⊗p (dv 3:p+2 )and[START_REF] Yau | Bayesian non-parametric hidden Markov models with applications in genomics[END_REF] and[START_REF] Zermelo | Die Berechnung der Turnier-Ergebnisse als ein Maximumproblem der Wahrscheinlichkeitsrechnung[END_REF],|Q p π h(v, x, η , η ) -Q p π h(v , x , η , η )| 2ν -1 (1 -ν) p-2 x2:p+1∈X p (K(x 2 , v, v 3 ) + K(x 2 , v , v 3 )) p+2 k=4 K(x k-1 , v k-1 , v k )π ⊗p (dv 3:p+2 ) 4ν -1 (1 -ν) p-2 .

  (X 1 , . . . , X -1 , x , X * +1 , . . . , X * n |F ] , b = sup x ∈X E[f (X 1 , . . . , X -1 , x , X * +1 , . . . , X * n |F ] .

γ

  s ∆(P s-) and b = b + n s= +1

  X 1 , . . . , X ) + [W ,s (X 1 , . . . , X , X * s+1 , . . . , X * n )|F ].Defining H = E π [f (X 0 , . . . , X , X * +1 , . . . , X * [W ,s (X 1 , . . . , X , X s+1 , . . . , X n )|F ] -E π [W ,s (X 1 , . . . , X , X * s+1 , . . . , X * n )|F ] .

	n-1
	E π
	s=

n )|F ] yields |G -H | n-1 s= E π

http://www.stats.ox.ac.uk/∼caron/code/bayesbt/index.html

By Markov inequality,

where the last inequality follows from Proposition 1 and [15, Lemma 10]. Choosing g : u → e -τ u 2 /2 with τ = (c 3 + c 1 )/(c/(2c) -1) 2 yields g(u n )e c3nε 2 n = e -c1nε 2 n /2 and by H2-(c 0 , c 1 , c 2 ),

Then, with

which concludes the proof.

B Forgetting properties of the loglikelihood Lemma 6. Assume that H1 holds. For any i 1, conditionally on

such that, for all 1 k i, there exists a measure µ k,i satisfying for all measurable set A:

where µ i,i = π and for

) .

On the other hand, for all k i + 1 and all measurable set A,

Proof. The proof is a classical result for strong mixing latent data models, it follows closely [START_REF] Douc | Asymptotic properties of the maximum likelihood estimator in autoregressive models with markov regime[END_REF].

See also [START_REF] Diel | Learning latent structure of large random graphs[END_REF] for a proof in the framework described by Figure 2.

Lemma 7. Assume that H1 holds. For all distribution π and all i 1, 0,

satisfy (42). By McDiarmid's inequality, for all λ > 0,

Then, by Markov's inequality, for any s > 0,

Applying this inequality with λ = 4t/D n concludes the proof.

Corollary 12. For any distribution π on X, any γ ∈ R n + , any f ∈ B(γ) and any t > 0,

where ν and η are given by (40).

For any x ∈ X, Q(x, •) is a probability distribution and

Therefore, for all (x, x ) ∈ X 2 ,

which yields, for all q 1, ∆(P q ) (∆(P )) q (1 -ν) q . By Theorem 11, for any distribution π on X, any γ ∈ R n + , any f ∈ B(γ) and any t > 0,

Note that by Cauchy-Schwarz inequality,

which concludes the proof.