A Moreau-Yosida approximation scheme for high-dimensional posterior and quasi-posterior distributions

Exact-sparsity inducing prior distributions in high-dimensional Bayesian analysis typically lead to posterior distributions that are very challenging to handle by standard Markov Chain Monte Carlo methods. We propose a methodology to derive a smooth approximation of such posterior distributions. 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 a factor $O(\sqrt{\gamma})$ of the true posterior distribution, where $\gamma>0$ is a user-controlled parameter that defines the approximation. We illustrate the method with a variable selection problem in linear regression models, and with a Gaussian graphical model selection problem.