Small-noise approximation for Bayesian optimal experimental design with nuisance uncertainty

TL;DR

This paper introduces small-noise approximation methods to efficiently compute expected information gain in Bayesian experimental design involving nuisance parameters, reducing computational costs while maintaining accuracy.

Contribution

It proposes two novel small-noise approximation methods, the Double-loop Monte Carlo and Monte Carlo Laplace, for handling nuisance uncertainty efficiently in Bayesian design.

Findings

Methods reduce computational complexity comparable to nuisance-free cases.

Demonstrated effectiveness on three examples, including PDE-based electrical impedance tomography.

Approximations maintain accuracy with lower computational cost.

Abstract

Calculating the expected information gain in optimal Bayesian experimental design typically relies on nested Monte Carlo sampling. When the model also contains nuisance parameters, which are parameters that contribute to the overall uncertainty of the system but are of no interest in the Bayesian design framework, this introduces a second inner loop. We propose and derive a small-noise approximation for this additional inner loop. The computational cost of our method can be further reduced by applying a Laplace approximation to the remaining inner loop. Thus, we present two methods, the small-noise Double-loop Monte Carlo and small-noise Monte Carlo Laplace methods. Moreover, we demonstrate that the total complexity of these two approaches remains comparable to the case without nuisance uncertainty. To assess the efficiency of these methods, we present three examples, and the last example includes the partial differential equation for the electrical impedance tomography experiment for composite laminate materials.