Statistiques bayésiennes semi-paramétriques – SP Bayes

The use of Bayesian statistics in complex models has grown dramatically due to the development of efficient implementation procedures such as the MCMC algorithms, which allow us to simulate the posterior distribution for almost any kind of models. This is particularly true for the use of Bayesian nonparametric or semi-parametric methods. In the last decade, the theoretical properties of Bayesian methods in infinite dimensional models have been studied, in particular their asymptotic properties, that is the consistency of the posterior and possibly the rate of convergence of the posterior. But, compared to the frequentist methods the knowledge of their asymptotic properties is still not advanced enough. The existing results mainly link conditions on the priors to rates of concentration of the posterior distribution toward the true parameter or to the existence of such a concentration. However major gaps remain in the theory, in particular for semi-parametric models. In this project we intend to study Bayesian semi-parametric inference, both from a theoretical and from a methodological point of view. Semi-parametric models can be separated into two main categories: regular models, for which the parameter of interest can be estimated with a typical parametric rate, i.e. the square root of the number of observations and non regular models for which the minimax rate of convergence for the parameter of interest is nonparametric. In this project we want to consider both categories. We want to study precise properties of Bayes estimators in regular models, such as the existence of Bernstein Von Mises theorems, so that Bayes procedures are asymptotically equivalent to likelihood procedures. Very few results exist on the existence of Bernstein Von Mises theorems, and the only existing results consider conjuguate types of priors so that the posterior is almost explicit. Our aim is to be able to link general conditions on the prior with the existence of Bernstein Von Mises theorems. We also want to study, in some non regular semi-parametric models the rate of convergence of the Bayes estimators to be able to compare these Bayesian procedures with minimax procedures and to understand better the impact of the prior. These studies follow existing results obtained by members of the project, either in a purely nonparametric Bayesian setting or in a frequentist setting. These two theoretical aspects of the project concern asymptotic properties of Bayesian procedures. It is also important to develop good procedures to implement these approaches. This methodological aspect will also be developed in our project, together with an application in a complex problem which will be conducted in collaboration with K. Mengersen (Australia) on quality control models for the water using satellite images.