biostudies-literatureUnknown78(4)Zhou XThis paper advocates proximal Markov Chain Monte Carlo (ProxMCMC) as a flexible and general Bayesian inference framework for constrained or regularized estimation. Originally introduced in the Bayesian imaging literature, ProxMCMC employs the Moreau-Yosida envelope for a smooth approximation of the total-variation regularization term, fixes variance and regularization strength parameters as constants, and uses the Langevin algorithm for the posterior sampling. We extend ProxMCMC to be fully Bayesian by providing data-adaptive estimation of all parameters including the regularization strength parameter. More powerful sampling algorithms such as Hamiltonian Monte Carlo are employed to scale ProxMCMC to high-dimensional problems. Analogous to the proximal algorithms in optimization, ProxMCMC The American statistician379-390https://www.ebi.ac.uk/biostudies/studies/S-EPMC11670895biostudies-literatureProximal MCMC for Bayesian Inference of Constrained and Regularized Estimation.PMC11670895Chi ECHeng QZhou XZhou HfalseProximal MCMC for Bayesian Inference of Constrained and Regularized Estimation.This paper advocates proximal Markov Chain Monte Carlo (ProxMCMC) as a flexible and general Bayesian inference framework for constrained or regularized estimation. Originally introduced in the Bayesian imaging literature, ProxMCMC employs the Moreau-Yosida envelope for a smooth approximation of the total-variation regularization term, fixes variance and regularization strength parameters as constants, and uses the Langevin algorithm for the posterior sampling. We extend ProxMCMC to be fully Bayesian by providing data-adaptive estimation of all parameters including the regularization strength parameter. More powerful sampling algorithms such as Hamiltonian Monte Carlo are employed to scale ProxMCMC to high-dimensional problems. Analogous to the proximal algorithms in optimization, ProxMCMC 2024-01-01T00:00:00Z20242025-04-18T22:01:22.822Z2025-04-07T09:56:54.291ZS-EPMC116708953972682610.1080/00031305.2024.2308821