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 offers a versatile and modularized procedure for conducting statistical inference on constrained and regularized problems. The power of ProxMCMC is illustrated on various statistical estimation and machine learning tasks, the inference of which is traditionally considered difficult from both frequentist and Bayesian perspectives.

翻译：本文推崇近端马尔可夫链蒙特卡洛方法（ProxMCMC）作为一种灵活通用的贝叶斯推断框架，适用于约束或正则化估计问题。该方法最初源于贝叶斯成像文献，通过Moreau-Yosida包络对总变分正则化项进行平滑逼近，将方差与正则化强度参数设为常数，并利用朗之万算法进行后验采样。我们通过提供所有参数（包括正则化强度参数）的数据自适应估计，将ProxMCMC扩展为完全贝叶斯方法。采用哈密顿蒙特卡洛等更强大的采样算法，使ProxMCMC能够扩展到高维问题。与优化领域的近端算法类似，ProxMCMC为约束与正则化问题的统计推断提供了一种灵活模块化的流程。我们通过多种传统上从频率学派和贝叶斯学派角度均被认为难以推断的统计估计与机器学习任务，展示了ProxMCMC的强大能力。