Hyperpriors for Mat\'ern fields with applications in Bayesian inversion

TL;DR

This paper develops non-stationary Matérn field priors with hyperpriors for Bayesian inversion, enabling better modeling of spatially varying correlation lengths and improved handling of smoothness and edges in inverse problems.

Contribution

It introduces a novel framework for non-stationary Matérn priors with hyperpriors, including discretisation and convergence analysis, applied to interpolation and differentiation tasks.

Findings

Bayesian inversion can effectively balance smoothness and edge-preservation.

Discretised models converge to the continuous prior and posterior.

Sampling algorithms efficiently compute the posterior mean.

Abstract

We introduce non-stationary Mat\'ern field priors with stochastic partial differential equations, and construct correlation length-scaling with hyperpriors. We model both the hyperprior and the Mat\'ern prior as continuous-parameter random fields. As hypermodels, we use Cauchy and Gaussian random fields, which we map suitably to a desired correlation length-scaling range. For computations, we discretise the models with finite difference methods. We consider the convergence of the discretised prior and posterior to the discretisation limit. We apply the developed methodology to certain interpolation and numerical differentiation problems, and show numerically that we can make Bayesian inversion which promotes competing constraints of smoothness and edge-preservation. For computing the conditional mean estimator of the posterior distribution, we use a combination of Gibbs and Metropolis-within-Gibbs sampling algorithms.