Well-posed Bayesian inverse problems and heavy-tailed stable quasi-Banach space priors

TL;DR

This paper extends Bayesian inverse problem frameworks to include heavy-tailed stable distribution priors in infinite-dimensional spaces, demonstrating sampling methods and posterior stability under weaker assumptions.

Contribution

It introduces a method to sample heavy-tailed stable priors in quasi-Banach spaces and proves Lipschitz continuity of the posterior with respect to data perturbations.

Findings

Sampling methods for stable distribution priors in quasi-Banach spaces.

Lipschitz continuity of the Bayesian posterior under weaker regularity conditions.

Extension of Bayesian inverse problem theory to heavy-tailed priors.

Abstract

This article extends the framework of Bayesian inverse problems in infinite-dimensional parameter spaces, as advocated by Stuart (Acta Numer. 19:451--559, 2010) and others, to the case of a heavy-tailed prior measure in the family of stable distributions, such as an infinite-dimensional Cauchy distribution, for which polynomial moments are infinite or undefined. It is shown that analogues of the Karhunen--Lo\`eve expansion for square-integrable random variables can be used to sample such measures on quasi-Banach spaces. Furthermore, under weaker regularity assumptions than those used to date, the Bayesian posterior measure is shown to depend Lipschitz continuously in the Hellinger metric upon perturbations of the misfit function and observed data.