ROUSSEAU Judith

In this paper, we consider nonparametric multidimensional finite mixture models and we are interested in the semiparametric estimation of the population weights. Here, the i.i.d. observations are assumed to have at least three components which are independent given the population. We approximate the semiparametric model by projecting the conditional distributions on step functions associated to some partition. Our first main result is that if we refine the partition slowly enough, the associated sequence of maximum likelihood estimators of the weights is asymptotically efficient, and the posterior distribution of the weights, when using a Bayesian procedure, satisfies a semiparametric Bernstein von Mises theorem. We then propose a cross-validation like procedure to select the partition in a finite horizon. Our second main result is that the proposed procedure satisfies an oracle inequality. Numerical experiments on simulated data illustrate our theoretical results.

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We consider a novel paradigm for Bayesian testing of hypotheses and Bayesian model comparison. Our alternative to the traditional construction of posterior probabilities that a given hypothesis is true or that the data originates from a specific model is to consider the models under comparison as components of a mixture model. We therefore replace the original testing problem with an estimation one that focus on the probability weight of a given model within a mixture model. We analyse the sensitivity on the resulting posterior distribution on the weights of various prior modelling on the weights. We stress that a major appeal in using this novel perspective is that generic improper priors are acceptable, while not putting convergence in jeopardy. Among other features, this allows for a resolution of the Lindley-Jeffreys paradox. When using a reference Beta B(a,a) prior on the mixture weights, we note that the sensitivity of the posterior estimations of the weights to the choice of a vanishes with the sample size increasing and advocate the default choice a=0.5, derived from Rousseau and Mengersen (2011). Another feature of this easily implemented alternative to the classical Bayesian solution is that the speeds of convergence of the posterior mean of the weight and of the corresponding posterior probability are quite similar.

Approximate Bayesian computation (ABC) is becoming an accepted tool for statistical analysis in models with intractable likelihoods. With the initial focus being primarily on the practical import of ABC, exploration of its formal statistical properties has begun to attract more attention. In this paper we consider the asymptotic behavior of the posterior obtained from ABC and the ensuing posterior mean. We give general results on: (i) the rate of concentration of the ABC posterior on sets containing the true parameter (vector). (ii) the limiting shape of the posterior. and\ (iii) the asymptotic distribution of the ABC posterior mean. These results hold under given rates for the tolerance used within ABC, mild regularity conditions on the summary statistics, and a condition linked to identification of the true parameters. Using simple illustrative examples that have featured in the literature, we demonstrate that the required identification condition is far from guaranteed. The implications of the theoretical results for practitioners of ABC are also highlighted.

Nonparametric Bayesian estimation for multidimensional Hawkes processes. SMAI Congress 2015.

Non parametric Bayesian estimation for Hawkes processes. International Society for Bayesian Analysis World Meeting, ISBA 2014.

We consider finite state space stationary hidden Markov models (HMMs) in the situation where the number of hidden states is unknown. We provide a frequentist asymptotic evaluation of Bayesian analysis methods. Our main result gives posterior concentration rates for the marginal densities, that is for the density of a fixed number of consecutive observations. Using conditions on the prior, we are then able to define a consistent Bayesian estimator of the number of hidden states. It is known that the likelihood ratio test statistic for overfitted HMMs has a non standard behaviour and is unbounded. Our conditions on the prior may be seen as a way to penalize parameters to avoid this phenomenon. Inference of parameters is a much more difficult task than inference of marginal densities, we still provide a precise description of the situation when the observations are i.i.d. and we allow for 2 possible hidden states.

Empirical Bayes methods are often thought of as a bridge between classical and Bayesian inference. In fact, in the literature the term empirical Bayes is used in quite diverse contexts and with different motivations. In this article, we provide a brief overview of empirical Bayes methods highlighting their scopes and meanings in different problems. We focus on recent results about merging of Bayes and empirical Bayes posterior distributions that regard popular, but otherwise debatable, empirical Bayes procedures as computationally convenient approximations of Bayesian solutions.

In this paper we consider non parametric finite translation mixtures. We prove that all the parameters of the model are identifiable as soon as the matrix that defines the joint distribution of two consecutive latent variables is non singular and the translation parameters are distinct. Under this assumption, we provide a consistent estimator of the number of populations, of the translation parameters and of the distribution of two consecutive latent variables, which we prove to be asymptotically normally distributed under mild dependency assumptions. We propose a non parametric estimator of the unknown translated density. In case the latent variables form a Markov chain (Hidden Markov models), we prove an oracle inequality leading to the fact that this estimator is minimax adaptive over regularity classes of densities.

The main part of this thesis concerns the asymptotic predictions of some confidence regions. It is organized around three axes: the first one concerns the study of Bayesian and frequentist covers of two types of confidence regions, the hpd regions and the joint bilateral intervals, when the observations are not discrete. In these two cases, asymptotic developments of the a posteriori and frequentist covers are obtained, as well as conditions of agreement between the two approaches. The second axis concerns the existence of asymptotic developments, at higher orders, of frequentist covers of some confidence regions in the discrete case. The third axis studies the effectiveness of continuity corrections on these asymptotic developments (still in the discrete case), determines asymptotic properties at higher orders of these rcorrected regions. For each of these three axes, the existence of nuisance parameters is considered. The last chapter deals with point estimation by studying the asymptotic bayes risk for a large family of loss functions.