Proximal Markov Chain Monte Carlo is a novel construct that lies at the intersection of Bayesian computation and convex optimization, which helped popularize the use of nondifferentiable priors in Bayesian statistics. Existing formulations of proximal MCMC, however, require hyperparameters and regularization parameters to be prespecified. In this work, we extend the paradigm of proximal MCMC through introducing a novel new class of nondifferentiable priors called epigraph priors. As a proof of concept, we place trend filtering, which was originally a nonparametric regression problem, in a parametric setting to provide a posterior median fit along with credible intervals as measures of uncertainty. The key idea is to replace the nonsmooth term in the posterior density with its Moreau-Yosida envelope, which enables the application of the gradient-based MCMC sampler Hamiltonian Monte Carlo. The proposed method identifies the appropriate amount of smoothing in a data-driven way, thereby automating regularization parameter selection. Compared with conventional proximal MCMC methods, our method is mostly tuning free, achieving simultaneous calibration of the mean, scale and regularization parameters in a fully Bayesian framework. Supplementary materials for this article are available online.

翻译：Proximal Markov Clack Monte Carlo是位于巴伊西亚计算和 convex优化交汇处的一个新建筑,它有助于在巴伊西亚统计中推广使用不可区别的前科。不过,现有的近似MCMC的配方要求预先说明超参数和正规化参数。在这项工作中,我们通过引入新型的不可区分的前科新类别,即传记前科,扩展了准超光谱MCMC的范范范范。作为概念的证明,我们将趋势过滤(最初是一个非参数回归问题)放在一个参数设置中,以提供一个与可靠间隔相匹配的后部中位,作为不确定性的衡量尺度。关键的想法是用其Moreau-Yosida封套件取代后端密度中的非单词。这使得基于梯度的MC采样员汉密尔密尔顿·蒙特卡洛得以应用。拟议的方法确定了以数据驱动方式实现平稳的适当数量,从而将规范化参数选择自动化。与传统的准光谱MC方法相比,我们的方法主要是对目前可使用的线级标准进行全面校准。