Bayesian Trend Filtering via Proximal Markov Chain Monte Carlo.

Area of Science:

• Bayesian statistics
• Convex optimization
• Computational statistics

Background:

• Proximal Markov Chain Monte Carlo (MCMC) integrates Bayesian computation and convex optimization.
• Existing proximal MCMC methods necessitate prespecified hyperparameters and regularization parameters.
• Nondifferentiable priors are increasingly utilized in Bayesian statistics.

Purpose of the Study:

• To extend proximal MCMC by introducing a novel class of nondifferentiable priors: epigraph priors.
• To develop a tuning-free proximal MCMC method for automated regularization parameter selection.
• To apply the novel method to trend filtering, moving it from nonparametric to a parametric setting.

Main Methods:

• Introduced epigraph priors as a new class of nondifferentiable priors.
• Utilized the Moreau-Yosida envelope to approximate nonsmooth posterior terms.
• Employed Hamiltonian Monte Carlo, a gradient-based MCMC sampler.
• Applied the framework to trend filtering for posterior median fitting and uncertainty quantification.

Main Results:

• The proposed method automates regularization parameter selection in a data-driven manner.
• The approach allows for simultaneous calibration of mean, scale, and regularization parameters within a Bayesian framework.
• The method demonstrated a tuning-free characteristic compared to conventional proximal MCMC techniques.
• Successfully provided posterior median fits and credible intervals for trend filtering.

Conclusions:

• The novel epigraph prior approach significantly advances proximal MCMC by reducing the need for manual parameter tuning.
• This work offers a robust and automated Bayesian framework for statistical modeling involving nondifferentiable priors.
• The method provides a powerful tool for analyzing data, particularly in settings like trend filtering, with built-in uncertainty estimation.