Designing truncated priors for direct and inverse Bayesian problems

TL;DR

This paper develops a theoretical framework for the asymptotic behavior of Bayesian inverse problems using truncated Gaussian priors, highlighting the importance of truncation points in both direct and inverse settings.

Contribution

It introduces a posterior contraction theory for linear inverse problems with truncated Gaussian priors under general smoothness conditions.

Findings

Established posterior contraction rates for truncated Gaussian priors.

Demonstrated the role of truncation points in inverse problem performance.

Linked direct and inverse problems through the modulus of continuity.

Abstract

The Bayesian approach to inverse problems with functional unknowns, has received significant attention in recent years. An important component of the developing theory is the study of the asymptotic performance of the posterior distribution in the frequentist setting. The present paper contributes to the area of Bayesian inverse problems by formulating a posterior contraction theory for linear inverse problems, with truncated Gaussian series priors, and under general smoothness assumptions. Emphasis is on the intrinsic role of the truncation point both for the direct as well as for the inverse problem, which are related through the modulus of continuity as this was recently highlighted by Knapik and Salomond (2018).