A Bernstein--von-Mises theorem for the Calder\'on problem with piecewise constant conductivities

TL;DR

This paper establishes a Bernstein--von-Mises theorem for the Calderón problem with piecewise constant conductivities, providing theoretical guarantees for Bayesian estimation in this inverse problem setting.

Contribution

It proves the invertibility of the information operator under injectivity conditions, leading to optimal Bayesian estimation guarantees for the problem.

Findings

Invertibility of the information operator is established.

Bernstein--von-Mises theorem is proved for the model.

Optimality guarantees for Bayesian posterior means are derived.

Abstract

This note considers a finite dimensional statistical model for the Calder\'on problem with piecewise constant conductivities. In this setting it is shown that injectivity of the forward map and its linearisation suffice to prove the invertibility of the information operator, resulting in a Bernstein--von-Mises theorem and optimality guarantees for estimation by Bayesian posterior means.