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

Abstract : The first main focus is the sparse Gaussian sequence model. An Empirical Bayes approach is used on the Spike and Slab prior to derive minimax convergence of the posterior second moment for Cauchy Slabs and a suboptimality result for the Laplace Slab is proved. Next, with a special choice of Slab convergence with the sharp minimax constant is derived. The second main focus is the density estimation model using a special Polya tree prior where the variables in the tree construction follow a Spike and Slab type distribution. Adaptive minimax convergence in the supremum norm of the posterior distribution as well as a nonparametric Bernstein-von Mises theorem are obtained
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Romain Mismer. Convergence et spike and Slab Bayesian posterior distributions in some high dimensional models. General Mathematics [math.GM]. Université Sorbonne Paris Cité, 2019. English. ⟨NNT : 2019USPCC064⟩. ⟨tel-02941474⟩

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