Variational Bayesian Optimal Experimental Design with Normalizing Flows

TL;DR

This paper introduces a novel variational Bayesian experimental design method using normalizing flows to efficiently estimate information gain without explicit likelihoods, demonstrated on complex models.

Contribution

The paper proposes vOED-NFs, integrating normalizing flows into variational OED to improve posterior approximation and reduce bias in EIG estimation.

Findings

Normalizing flows improve EIG estimation accuracy.

vOED-NFs captures complex, multi-modal posteriors effectively.

Method outperforms previous approaches in benchmark tests.

Abstract

Bayesian optimal experimental design (OED) seeks experiments that maximize the expected information gain (EIG) in model parameters. Directly estimating the EIG using nested Monte Carlo is computationally expensive and requires an explicit likelihood. Variational OED (vOED), in contrast, estimates a lower bound of the EIG without likelihood evaluations by approximating the posterior distributions with variational forms, and then tightens the bound by optimizing its variational parameters. We introduce the use of normalizing flows (NFs) for representing variational distributions in vOED; we call this approach vOED-NFs. Specifically, we adopt NFs with a conditional invertible neural network architecture built from compositions of coupling layers, and enhanced with a summary network for data dimension reduction. We present Monte Carlo estimators to the lower bound along with gradient expressions to enable a gradient-based simultaneous optimization of the variational parameters and the design variables. The vOED-NFs algorithm is then validated in two benchmark problems, and demonstrated on a partial differential equation-governed application of cathodic electrophoretic deposition and an implicit likelihood case with stochastic modeling of aphid population. The findings suggest that a composition of 4--5 coupling layers is able to achieve lower EIG estimation bias, under a fixed budget of forward model runs, compared to previous approaches. The resulting NFs produce approximate posteriors that agree well with the true posteriors, able to capture non-Gaussian and multi-modal features effectively.