A Proximal Markov Chain Monte Carlo Method for Bayesian Inference in Imaging Inverse Problems: When Langevin Meets Moreau

Alain Durmus, Éric Moulines, Marcelo Pereyra

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)


Modern imaging methods rely strongly on Bayesian inference techniques to solve challenging imaging problems. Currently, the predominant Bayesian computational approach is convex optimization, which scales very efficiently to high-dimensional image models and delivers accurate point estimation results. However, in order to perform more complex analyses, for example, image uncertainty quantification or model selection, it is often necessary to use more computationally intensive Bayesian computation techniques such as Markov chain Monte Carlo methods. This paper presents a new and highly efficient Markov chain Monte Carlo methodology to perform Bayesian computation for high-dimensional models that are log-concave and nonsmooth, a class of models that is central in imaging sciences. The methodology is based on a regularized unadjusted Langevin algorithm that exploits tools from convex analysis, namely, Moreau-Yosida envelopes and proximal operators, to construct Markov chains with favorable convergence properties. In addition to scaling efficiently to high dimensions, the method can be applied in a straightforward manner to models that are currently solved using proximal optimization algorithms. We provide a detailed theoretical analysis of the proposed methodology, including asymptotic and nonasymptotic convergence results with easily verifiable conditions, and explicit bounds on the convergence rates. The proposed methodology is demonstrated with five experiments related to image deconvolution and tomographic reconstruction with total-variation and \ell 1 priors, where we conduct a range of challenging Bayesian analyses related to uncertainty quantification, hypothesis testing, and model selection in the absence of ground truth.

Original languageEnglish
Pages (from-to)991-1028
Number of pages38
JournalSIAM Review
Issue number4
Early online date3 Nov 2022
Publication statusPublished - 2022


  • Bayesian inference
  • convex optimization
  • inverse problems
  • Markov chain Monte Carlo methods
  • mathematical imaging
  • model selection
  • uncertainty quantification

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computational Mathematics
  • Applied Mathematics


Dive into the research topics of 'A Proximal Markov Chain Monte Carlo Method for Bayesian Inference in Imaging Inverse Problems: When Langevin Meets Moreau'. Together they form a unique fingerprint.

Cite this