Bayesian Trend Filtering via Proximal Markov Chain Monte Carlo

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.

近端马尔可夫链蒙特卡洛（Proximal Markov Chain Monte Carlo）是一种新兴研究框架，坐落于贝叶斯计算与凸优化的交叉领域，其推动了不可微先验在贝叶斯统计中的应用普及。然而，现有近端马尔可夫链蒙特卡洛方法均要求预先指定超参数与正则化参数。在本研究中，我们通过引入一类全新的不可微先验——上图先验（epigraph priors），拓展了近端马尔可夫链蒙特卡洛的研究范式。作为概念验证，我们将原本属于非参数回归问题的趋势滤波置于参数化框架之中，以提供后验中位数拟合结果与作为不确定性度量的可信区间。核心思路是将后验密度中的非光滑项替换为其莫罗-约西达包络（Moreau-Yosida envelope），从而得以应用基于梯度的马尔可夫链蒙特卡洛采样器——哈密顿蒙特卡洛。所提方法以数据驱动的方式确定合适的平滑程度，实现了正则化参数选择的自动化。与传统近端马尔可夫链蒙特卡洛方法相比，本方法基本无需调参，可在全贝叶斯框架下同时对均值、尺度与正则化参数进行校准。本文的补充材料可在线获取。