Well-posedness of Bayesian inverse problems in quasi-Banach spaces with stable priors

TL;DR

This paper extends the well-posedness framework of Bayesian inverse problems to infinite-dimensional quasi-Banach spaces with stable priors, demonstrating continuous dependence of the posterior on data and perturbations, even with heavy-tailed distributions.

Contribution

It introduces a novel approach for sampling stable priors in quasi-Banach spaces and proves Lipschitz continuity of the Bayesian posterior under weaker regularity conditions.

Findings

Extension of well-posed BIPs to quasi-Banach spaces with stable priors.

Sampling methods for stable measures using Karhunen--Loève type expansions.

Posterior measure depends Lipschitz continuously on data and model perturbations.

Abstract

The Bayesian perspective on inverse problems has attracted much mathematical attention in recent years. Particular attention has been paid to Bayesian inverse problems (BIPs) in which the parameter to be inferred lies in an infinite-dimensional space, a typical example being a scalar or tensor field coupled to some observed data via an ODE or PDE. This article gives an introduction to the framework of well-posed BIPs in infinite-dimensional parameter spaces, as advocated by Stuart (Acta Numer. 19:451--559, 2010) and others. This framework has the advantage of ensuring uniformly well-posed inference problems independently of the finite-dimensional discretisation used for numerical solution. Recently, this framework has been extended 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 and total variation metrics upon perturbations of the misfit function and observed data.