Efficient Bayesian Computation by Proximal Markov Chain Monte Carlo: When Langevin Meets Moreau.

Alain Durmus (1, 2) , Éric Moulines (1, 3, 4) , Marcelo Pereyra (5)

1 LTCI - Laboratoire Traitement et Communication de l'Information
2 CMLA - Centre de Mathématiques et de Leurs Applications
3 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
4 XPOP - Modélisation en pharmacologie de population
5 School of Mathematics [Bristol]

In this paper, two new algorithms to sample from possibly non-smooth log-concave probability measures are introduced. These algorithms use Moreau-Yosida envelope combined with the Euler-Maruyama discretization of Langevin diffusions. They are applied to a de-convolution problem in image processing, which shows that they can be practically used in a high dimensional setting. Finally, non-asymptotic bounds for one of the proposed methods are derived. These bounds follow from non-asymptotic results for ULA applied to probability measures with a convex continuously differentiable log-density with respect to the Lebesgue measure.

Mots clés: Markov Chain Monte Carlo; Rate of convergence; Metropolis Adjusted Langevin Algorithm

Domaines: Méthodologie [stat.ME]; Théorie [stat.TH]

Alain Durmus, Éric Moulines, Marcelo Pereyra. Efficient Bayesian Computation by Proximal Markov Chain Monte Carlo: When Langevin Meets Moreau.. SIAM Journal on Imaging Sciences, 2018, 11 (1), ⟨10.1137/16M110834⟩. ⟨hal-01267115⟩