Bayesian inverse problems with non-commuting operators

TL;DR

This paper extends Bayesian inverse problem analysis in Hilbert spaces to cases where the key operators do not commute, using a link condition to derive bounds on the posterior distribution's contraction.

Contribution

It introduces a framework for handling non-commuting operators in Bayesian inverse problems, replacing the commutativity assumption with a link condition, enabling new analysis techniques.

Findings

Established a link condition for non-commuting operators

Derived bounds for posterior contraction rates

Connected non-commuting case to the commuting case via interpolation

Abstract

The Bayesian approach to ill-posed operator equations in Hilbert space recently gained attraction. In this context, and when the prior distribution is Gaussian, then two operators play a significant role, the one which governs the operator equation, and the one which describes the prior covariance. Typically it is assumed that these operators commute. Here we extend this analysis to non-commuting operators, replacing the commutativity assumption by a link condition. We discuss its relation to the commuting case, and we indicate that this allows to use interpolation type results to obtain tight bounds for the contraction of the posterior Gaussian distribution towards the data generating element.