On Bayes risk of the posterior mean in linear inverse problems

TL;DR

This paper revisits the Bayes risk of the posterior mean in finite-dimensional ill-posed linear inverse problems with Gaussian models, showing it equals the trace of the posterior covariance, and discusses implications for infinite-dimensional cases.

Contribution

It provides a clear derivation of the Bayes risk equaling the trace of the posterior covariance in Gaussian linear inverse problems, linking finite and infinite-dimensional frameworks.

Findings

Bayes risk of the posterior mean equals the trace of the posterior covariance.

The derivation is extended from finite to motivated infinite-dimensional settings.

Highlights the relationship between estimation error and posterior covariance in Gaussian models.

Abstract

We consider the concept of Bayes risk in the context of finite-dimensional ill-posed linear inverse problem with Gaussian prior and noise models. In this note, we rederive the following well-known result: in the present Gaussian linear setting, the Bayes risk of the posterior mean, relative to the sum of squares loss function, equals the trace of the posterior covariance operator. While our discussion is limited to a finite-dimensional setup, our presentation is motivated by more general infinite-dimensional formulations.