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