Well-posed Bayesian Inverse Problems: Priors with Exponential Tails

TL;DR

This paper investigates the conditions under which Bayesian inverse problems with priors having exponential tails are well-posed, ensuring existence, uniqueness, and stability of the posterior in infinite-dimensional spaces.

Contribution

It provides criteria for well-posedness with convex, log-concave priors including Gaussian and Besov measures, and offers a method to construct such priors on Banach spaces.

Findings

Established conditions for existence and stability of the posterior

Analyzed discretization and approximation methods for the posterior

Presented a construction recipe for convex priors on Banach spaces

Abstract

We consider the well-posedness of Bayesian inverse problems when the prior measure has exponential tails. In particular, we consider the class of convex (log-concave) probability measures which include the Gaussian and Besov measures as well as certain classes of hierarchical priors. We identify appropriate conditions on the likelihood distribution and the prior measure which guarantee existence, uniqueness and stability of the posterior measure with respect to perturbations of the data. We also consider consistent approximations of the posterior such as discretization by projection. Finally, we present a general recipe for construction of convex priors on Banach spaces which will be of interest in practical applications where one often works with spaces such as $L^2$ or the continuous functions.