Convergence et spike and Slab Bayesian posterior distributions in some high dimensional models.

Summary We first focus on the parsimonious Gaussian sequence model. An empirical Bayesian approach on the a priori Spike and Slab allows us to obtain the convergence at minimax speed of the second order moment a posteriori for Cauchy Slabs and we prove a suboptimality result for a Laplace Slab. A better choice of Slab allows us to obtain the exact constant. In the density estimation model, an a priori Polya tree such that the variables in the tree have a Spike and Slab distribution gives minimax and adaptive speed convergence for the sup norm of the a posteriori law and a nonparametric Bernstein-von Mises theorem.