Bayesian Trend Filtering via Proximal Markov Chain Monte Carlo
Bayesian Trend Filtering via Proximal Markov Chain Monte Carlo
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DOI：
10.1080/10618600.2023.2170089
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发表时间：
2022-01
影响因子：
2.4
通讯作者：
Qiang Heng;Hua Zhou;Eric C. Chi
中科院分区：
文献类型：
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作者：
Qiang Heng;Hua Zhou;Eric C. Chi
近端马尔可夫链蒙特卡罗算法是贝叶斯计算与凸优化相结合的一种新构造，它有助于推广不可微先验在贝叶斯统计中的应用。然而，现有的近端MCMC公式需要预先指定超参数和正则化参数。在本文中，我们通过引入一类新的不可微先验，即铭文先验，扩展了近端MCMC的范式。作为概念证明，我们将趋势过滤（最初是一个非参数回归问题）置于参数设置中，以提供后验中位数拟合以及可信区间作为不确定性的度量。关键思想是将后验密度中的非光滑项替换为Moreau-Yosida包络，从而实现基于梯度的MCMC采样器哈密顿蒙特卡罗的应用。该方法以数据驱动的方式识别适当的平滑量，从而实现正则化参数选择的自动化。与传统的近端MCMC方法相比，我们的方法可以在完全贝叶斯框架下实现均值、尺度和正则化参数的同步校准。本文的补充材料可在网上获得。
Abstract 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 article, 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.