Laplace-based strategies for Bayesian optimal experimental design with nuisance uncertainty

TL;DR

This paper introduces two novel Laplace-based estimators to efficiently compute the expected information gain in Bayesian experimental design involving nuisance parameters, significantly reducing computational costs.

Contribution

The paper proposes two new Laplace approximation-based estimators for Bayesian experimental design with nuisance uncertainty, improving computational efficiency.

Findings

Both estimators effectively reduce computational effort.

Numerical examples demonstrate the estimators' applicability and accuracy.

The second estimator offers a bias correction over the first.

Abstract

Finding the optimal design of experiments in the Bayesian setting typically requires estimation and optimization of the expected information gain functional. This functional consists of one outer and one inner integral, separated by the logarithm function applied to the inner integral. When the mathematical model of the experiment contains uncertainty about the parameters of interest and nuisance uncertainty, (i.e., uncertainty about parameters that affect the model but are not themselves of interest to the experimenter), two inner integrals must be estimated. Thus, the already considerable computational effort required to determine good approximations of the expected information gain is increased further. The Laplace approximation has been applied successfully in the context of experimental design in various ways, and we propose two novel estimators featuring the Laplace approximation to alleviate the computational burden of both inner integrals considerably. The first estimator applies Laplace's method followed by a Laplace approximation, introducing a bias. The second estimator uses two Laplace approximations as importance sampling measures for Monte Carlo approximations of the inner integrals. Both estimators use Monte Carlo approximation for the remaining outer integral estimation. We provide four numerical examples demonstrating the applicability and effectiveness of our proposed estimators.