A Multi-fidelity Estimator of the Expected Information Gain for Bayesian Optimal Experimental Design

Optimal experimental design (OED) is a framework that leverages a mathematical model of the experiment to identify optimal conditions for conducting the experiment. Under a Bayesian approach, the design objective function is typically chosen to be the expected information gain (EIG). However, EIG is intractable for nonlinear models and must be estimated numerically. Estimating the EIG generally entails some variant of Monte Carlo sampling, requiring repeated data model and likelihood evaluations $\unicode{x2013}$ each involving solving the governing equations of the experimental physics $\unicode{x2013}$ under different sample realizations. This computation becomes impractical for high-fidelity models. We introduce a novel multi-fidelity EIG (MF-EIG) estimator under the approximate control variate (ACV) framework. This estimator is unbiased with respect to the high-fidelity mean, and minimizes variance under a given computational budget. We achieve this by first reparameterizing the EIG so that its expectations are independent of the data models, a requirement for compatibility with ACV. We then provide specific examples under different data model forms, as well as practical enhancements of sample size optimization and sample reuse techniques. We demonstrate the MF-EIG estimator in two numerical examples: a nonlinear benchmark and a turbulent flow problem involving the calibration of shear-stress transport turbulence closure model parameters within the Reynolds-averaged Navier-Stokes model. We validate the estimator’s unbiasedness and observe one- to two-orders-of-magnitude variance reduction compared to existing single-fidelity EIG estimators.

💡 Research Summary

The paper addresses a fundamental bottleneck in Bayesian optimal experimental design (OED): the expected information gain (EIG) is analytically intractable for nonlinear, high‑dimensional models and must be estimated via Monte‑Carlo (MC) methods. The standard nested MC (NMC) estimator requires repeated evaluations of the forward model (often a costly PDE solver) for each outer and inner sample, leading to prohibitive computational cost when high‑fidelity (HF) physics are involved.

To overcome this, the authors develop a multi‑fidelity EIG estimator (MF‑EIG) within the Approximate Control Variates (ACV) framework. The key insight is a re‑parameterization of the EIG that separates the expectation from the data‑model (likelihood) itself, yielding an expression of the form
(U(\xi)=\mathbb{E}_{y,\theta|\xi}\big