A brief note on the Bayesian D-optimality criterion

TL;DR

This paper provides a straightforward derivation showing that in finite-dimensional Bayesian linear inverse problems with Gaussian assumptions, the D-optimal design criterion aligns with minimizing the posterior covariance's log-determinant, aiding experimental design.

Contribution

It offers a simple, generic derivation of the equivalence between Bayesian D-optimality and minimizing the posterior covariance determinant, applicable to finite and potentially infinite-dimensional problems.

Findings

Derivation confirms the equivalence in finite-dimensional cases.

Framework is adaptable to infinite-dimensional inverse problems.

Clarifies the mathematical foundation of Bayesian D-optimality.

Abstract

We consider finite-dimensional Bayesian linear inverse problems with Gaussian priors and additive Gaussian noise models. The goal of this note is to present a simple derivation of the well-known fact that solving the Bayesian D-optimal experimental design problem, i.e., maximizing the expected information gain, is equivalent to minimizing the log-determinant of posterior covariance operator. We focus on finite-dimensional inverse problems. However, the presentation is kept generic to facilitate extensions to infinite-dimensional inverse problems.