On submodularity of the expected information gain

TL;DR

This paper proves that the expected information gain in finite-dimensional linear Gaussian Bayesian inverse problems is submodular, providing a simpler proof tailored to discretized PDE-constrained problems.

Contribution

It offers a straightforward alternative proof of submodularity for expected information gain in PDE-constrained inverse problems.

Findings

Expected information gain is submodular in the specified setting.

The proof is simplified and tailored to discretized PDE problems.

Supports efficient sensor placement strategies.

Abstract

We consider finite-dimensional linear Gaussian Bayesian inverse problems with uncorrelated sensor measurements. In this setting, it is known that the expected information gain, quantified by the expected Kullback-Leibler divergence from the posterior measure to the prior measure, is submodular. We present a simple alternative proof of this fact tailored to a weighted inner product space setting arising from discretization of infinite-dimensional inverse problems constrained by partial differential equations (PDEs).