On Bayesian A- and D-optimal experimental designs in infinite dimensions

TL;DR

This paper extends Bayesian A- and D-optimal experimental design concepts to infinite-dimensional Hilbert spaces, deriving formulas for information gain and establishing connections with the MAP estimator's Bayes risk.

Contribution

It introduces the infinite-dimensional versions of Bayesian D- and A-optimality, including formulas for information gain and the relation to the MAP estimator's Bayes risk.

Findings

Derived the infinite-dimensional Kullback-Leibler divergence expression.

Extended Bayesian A-optimality to infinite dimensions.

Established the equivalence between Bayes risk and posterior covariance trace in infinite dimensions.

Abstract

We consider Bayesian linear inverse problems in infinite-dimensional separable Hilbert spaces, with a Gaussian prior measure and additive Gaussian noise model, and provide an extension of the concept of Bayesian D-optimality to the infinite-dimensional case. To this end, we derive the infinite-dimensional version of the expression for the Kullback-Leibler divergence from the posterior measure to the prior measure, which is subsequently used to derive the expression for the expected information gain. We also study the notion of Bayesian A-optimality in the infinite-dimensional setting, and extend the well known (in the finite-dimensional case) equivalence of the Bayes risk of the MAP estimator with the trace of the posterior covariance, for the Gaussian linear case, to the infinite-dimensional Hilbert space case.