# Efficient Bayesian computation by proximal Markov chain Monte Carlo: when Langevin meets Moreau

arXiv: https://arxiv.org/abs/1612.07471 · HTML: https://mlsanity.com/paper/1612.07471/

## Key facts

- Cited by 84 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Dec 2016.

## Built on by (most cited first, 30 of 84)

- [Denoising Diffusion Models for Plug-and-Play Image Restoration](https://mlsanity.com/paper/2305.08995/) (2023)
- [Bayesian imaging using Plug & Play priors: when Langevin meets Tweedie](https://mlsanity.com/paper/2103.04715/) (2021)
- [Improved dimension dependence of a proximal algorithm for sampling](https://mlsanity.com/paper/2302.10081/) (2023)
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- [Bayesian Imaging With Data-Driven Priors Encoded by Neural Networks: Theory, Methods, and Algorithms](https://mlsanity.com/paper/2103.10182/) (2021)
- [A Proximal Algorithm for Sampling from Non-smooth Potentials](https://mlsanity.com/paper/2110.04597/) (2021)
- [A Proximal Algorithm for Sampling](https://mlsanity.com/paper/2202.13975/) (2022)
- [Resolving the Mixing Time of the Langevin Algorithm to its Stationary Distribution for Log-Concave Sampling](https://mlsanity.com/paper/2210.08448/) (2022)
- [A Convergence Theory for SVGD in the Population Limit under Talagrand's Inequality T1](https://mlsanity.com/paper/2106.03076/) (2021)
- [Stable Deep MRI Reconstruction using Generative Priors](https://mlsanity.com/paper/2210.13834/) (2022)
- [Sampling with Mirrored Stein Operators](https://mlsanity.com/paper/2106.12506/) (2021)
- [Convergence of Stein Variational Gradient Descent under a Weaker Smoothness Condition](https://mlsanity.com/paper/2206.00508/) (2022)
- [Accelerating Neural Field Training via Soft Mining](https://mlsanity.com/paper/2312.00075/) (2023)
- [On Maximum-a-Posteriori estimation with Plug & Play priors and stochastic gradient descent](https://mlsanity.com/paper/2201.06133/) (2022)
- [Posterior-Variance-Based Error Quantification for Inverse Problems in Imaging](https://mlsanity.com/paper/2212.12499/) (2022)
- [Kinetic Langevin MCMC Sampling Without Gradient Lipschitz Continuity -- the Strongly Convex Case](https://mlsanity.com/paper/2301.08039/) (2023)
- [Asymptotic bias of inexact Markov Chain Monte Carlo methods in high dimension](https://mlsanity.com/paper/2108.00682/) (2021)
- [The split Gibbs sampler revisited: improvements to its algorithmic structure and augmented target distribution](https://mlsanity.com/paper/2206.13894/) (2022)
- [Non-asymptotic estimates for TUSLA algorithm for non-convex learning with applications to neural networks with ReLU activation function](https://mlsanity.com/paper/2107.08649/) (2021)
- [Langevin Monte Carlo for strongly log-concave distributions: Randomized midpoint revisited](https://mlsanity.com/paper/2306.08494/) (2023)
- [Regularization by denoising: Bayesian model and Langevin-within-split Gibbs sampling](https://mlsanity.com/paper/2402.12292/) (2024)
- [Parallelized Midpoint Randomization for Langevin Monte Carlo](https://mlsanity.com/paper/2402.14434/) (2024)
- [Plug-and-Play Posterior Sampling under Mismatched Measurement and Prior Models](https://mlsanity.com/paper/2310.03546/) (2023)
- [Task-Driven Uncertainty Quantification in Inverse Problems via Conformal Prediction](https://mlsanity.com/paper/2405.18527/) (2024)
- [Proximal Interacting Particle Langevin Algorithms](https://mlsanity.com/paper/2406.14292/) (2024)
- [Bregman Proximal Langevin Monte Carlo via Bregman--Moreau Envelopes](https://mlsanity.com/paper/2207.04387/) (2022)
- [Proximal Oracles for Optimization and Sampling](https://mlsanity.com/paper/2404.02239/) (2024)
- [Do Bayesian imaging methods report trustworthy probabilities?](https://mlsanity.com/paper/2405.08179/) (2024)
- [Empirical Bayesian image restoration by Langevin sampling with a denoising diffusion implicit prior](https://mlsanity.com/paper/2409.04384/) (2024)
- [Diffusion at Absolute Zero: Langevin Sampling Using Successive Moreau Envelopes [conference paper]](https://mlsanity.com/paper/2502.01358/) (2025)

## Abstract (arXiv)

Modern imaging methods rely strongly on Bayesian inference techniques to solve challenging imaging problems. Currently, the predominant Bayesian computation approach is convex optimisation, 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 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 non-smooth, a class of models that is central in imaging sciences. The methodology is based on a regularised unadjusted Langevin algorithm that exploits tools from convex analysis, namely Moreau-Yoshida envelopes and proximal operators, to construct Markov chains with favourable convergence properties. In addition to scaling efficiently to high dimensions, the method is straightforward to apply to models that are currently solved by using proximal optimisation algorithms. We provide a detailed theoretical analysis of the proposed methodology, including asymptotic and non-asymptotic convergence results with easily verifiable conditions, and explicit bounds on the convergence rates. The proposed methodology is demonstrated with four experiments related to image deconvolution and tomographic reconstruction with total-variation and ℓ1 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.