A Moreau-Yosida approximation scheme for a class of high-dimensional posterior distributions

Abstract

Exact-sparsity inducing prior distributions in Bayesian analysis typically lead to posterior distributions that are very challenging to handle by standard Markov Chain Monte Carlo (MCMC) methods, particular in high-dimensional models with large number of parameters. We propose a methodology to derive smooth approximations of such posterior distributions that are, in some cases, easier to handle by standard MCMC methods. The approximation is obtained from the forward-backward approximation of the Moreau-Yosida regularization of the negative log-density. We show that the derived approximation is within $O(\sqrt{\gamma })$ of the true posterior distribution in the $\beta$-metric, where $\gamma >0$ is a user-controlled parameter that defines the approximation. We illustrate the method with a high-dimensional linear regression model.

35 pages, 5 Figures

References in corpus (4)

• Nearly unbiased variable selection under minimax concave penalty
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• Forward-backward truncated Newton methods for convex composite optimization

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