On some recent advances in high dimensional Bayesian Statistics

This paper proposes to review some recent developments in Bayesian statistics for high dimensional data. After giving some brief motivations in a short introduction, we describe new advances in the understanding of Bayes posterior computation as well as theoretical contributions in non parametric and high dimensional Bayesian approaches. From an applied point of view, we describe the so-called SQMC particle method to compute posterior Bayesian law, and provide a nonparametric analysis of the widespread ABC method. On the theoretical side, we describe some recent advances in Bayesian consistency for a nonparametric hidden Markov model as well as new PAC-Bayesian results for different models of high dimensional regression.