Bayesian linear inverse problems in regularity scales

TL;DR

This paper establishes theoretical rates for the convergence of Bayesian posterior distributions in inverse problems across smoothness scales, emphasizing the role of prior concentration and Galerkin approximation.

Contribution

It provides a general framework for analyzing posterior contraction rates in inverse problems using abstract priors and Galerkin methods, applicable to various prior types including Gaussian and mixtures.

Findings

Near optimal and adaptive recovery with non-conjugate series priors.

Gaussian and mixture priors achieve near optimal contraction rates.

Applicable to inverse problems from differential equations.

Abstract

We obtain rates of contraction of posterior distributions in inverse problems defined by scales of smoothness classes. We derive abstract results for general priors, with contraction rates determined by Galerkin approximation. The rate depends on the amount of prior concentration near the true function and the prior mass of functions with inferior Galerkin approximation. We apply the general result to non-conjugate series priors, showing that these priors give near optimal and adaptive recovery in some generality, Gaussian priors, and mixtures of Gaussian priors, where the latter are also shown to be near optimal and adaptive. The proofs are based on general testing and approximation arguments, without explicit calculations on the posterior distribution. We are thus not restricted to priors based on the singular value decomposition of the operator. We illustrate the results with examples of inverse problems resulting from differential equations.