Dominique Bontemps

Bernstein–von Mises theorems for Gaussian regression with increasing number of regressors

This paper brings a contribution to the Bayesian theory of nonparametric and semiparametric estimation. We are interested in the asymptotic normality of the posterior distribution in Gaussian linear regression models when the number of regressors increases with the sample size. Two kinds of Bernstein-von Mises Theorems are obtained in this framework: nonparametric theorems for the parameter itself, and semiparametric theorems for functionals of the parameter. We apply them to the Gaussian sequence model and to the regression of functions in Sobolev and

Universal Coding on Infinite Alphabets: Exponentially Decreasing Envelopes

C^{\alpha}

About Adaptive Coding on Countable Alphabets

classes, in which we get the minimax convergence rates. Adaptivity is reached for the Bayesian estimators of functionals in our applications.

Bayesian methods for the Shape Invariant Model

This paper deals with the problem of universal lossless coding on a countable infinite alphabet. It focuses on some classes of sources defined by an envelope condition on the marginal distribution, namely exponentially decreasing envelope classes with exponent . The minimax redundancy of exponentially decreasing envelope classes is proved to be equivalent to . Then, an adaptive algorithm is proposed, whose maximum redundancy is equivalent to the minimax redundancy.

Clustering and variable selection for categorical multivariate data

This paper sheds light on adaptive coding with respect to classes of memoryless sources over a countable alphabet defined by an envelope function with finite and non-decreasing hazard rate (log-concave envelope distributions). We prove that the auto-censuring (AC) code is adaptive with respect to the collection of such classes. The analysis builds on the tight characterization of universal redundancy rate in terms of metric entropy and on a careful analysis of the performance of the AC-coding algorithm. The latter relies on nonasymptotic bounds for maxima of samples from discrete distributions with finite and nondecreasing hazard rate.

Posterior contraction rates on probability measures

In this paper, we consider the so-called Shape Invariant Model that is used to model a function f submitted to a random translation of law g in a white noise. This model is of interest when the law of the deformations is unknown. Our objective is to recover the law of the process Pf0,g0 as well as f and g. To do this, we adopt a Bayesian point of view and find priors on f and g so that the posterior distribution concentrates at a polynomial rate around Pf0,g0 when n goes to +∞. We then derive results on the identifiability of the SIM, as well as results on the functional objects themselves. We intensively use Bayesian non-parametric tools coupled with mixture models, which may be of independent interest in model selection from a frequentist point of view. AMS 2000 subject classifications: Primary 62G05, 62F15; secondary 62G20.