Cauchy Markov Random Field Priors for Bayesian Inversion

TL;DR

This paper reviews and introduces new Cauchy Markov random field priors for Bayesian inverse problems, comparing their properties and demonstrating their application using advanced MCMC methods in deconvolution tasks.

Contribution

It proposes new Cauchy difference priors, including isotropic and PDE-based variants, and compares them with existing priors using sophisticated MCMC techniques.

Findings

New Cauchy priors improve modeling flexibility.

Advanced MCMC methods effectively sample complex posteriors.

Numerical results demonstrate the priors' applicability in deconvolution.

Abstract

The use of Cauchy Markov random field priors in statistical inverse problems can potentially lead to posterior distributions which are non-Gaussian, high-dimensional, multimodal and heavy-tailed. In order to use such priors successfully, sophisticated optimization and Markov chain Monte Carlo (MCMC) methods are usually required. In this paper, our focus is largely on reviewing recently developed Cauchy difference priors, while introducing interesting new variants, whilst providing a comparison. We firstly propose a one-dimensional second order Cauchy difference prior, and construct new first and second order two-dimensional isotropic Cauchy difference priors. Another new Cauchy prior is based on the stochastic partial differential equation approach, derived from Mat\'{e}rn type Gaussian presentation. The comparison also includes Cauchy sheets. Our numerical computations are based on both maximum a posteriori and conditional mean estimation.We exploit state-of-the-art MCMC methodologies such as Metropolis-within-Gibbs, Repelling-Attracting Metropolis, and No-U-Turn sampler variant of Hamiltonian Monte Carlo. We demonstrate the models and methods constructed for one-dimensional and two-dimensional deconvolution problems. Thorough MCMC statistics are provided for all test cases, including potential scale reduction factors.