Cauchy Markov Random Field Priors for Bayesian Inversion

by   Neil K. Chada, et al.

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 Matérn 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.


page 20

page 21

page 22


Sequential pCN-MCMC, an efficient MCMC method for Bayesian inversion of high-dimensional multi-Gaussian priors

In geostatistics, Gaussian random fields are often used to model heterog...

A Two-Dimensional Intrinsic Gaussian Markov Random Field for Blood Pressure Data

Many real-world phenomena are naturally bivariate. This includes blood p...

Enhancing Industrial X-ray Tomography by Data-Centric Statistical Methods

X-ray tomography has applications in various industrial fields such as s...

High-dimensional Stochastic Inversion via Adjoint Models and Machine Learning

Performing stochastic inversion on a computationally expensive forward s...

Accelerating Metropolis-within-Gibbs sampler with localized computations of differential equations

Inverse problem is ubiquitous in science and engineering, and Bayesian m...

Bayesian Sparse Blind Deconvolution Using MCMC Methods Based on Normal-Inverse-Gamma Prior

Bayesian estimation methods for sparse blind deconvolution problems conv...

A Method for Compressing Parameters in Bayesian Models with Application to Logistic Sequence Prediction Models

Bayesian classification and regression with high order interactions is l...