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

2025-01-18

AI Summary

In nonlinear Bayesian optimal experimental design, the expected information gain (EIG) lacks closed-form solutions, while high-fidelity numerical estimation incurs prohibitive computational cost. To address this, we propose a multi-fidelity, unbiased EIG estimator based on approximate control variates (ACV). We first reformulate EIG via reparameterization to ensure unbiased mean estimation of the high-fidelity model within the ACV framework. Integrating sample reuse and adaptive sample allocation under a fixed budget, our method minimizes estimator variance. The approach unifies multi-fidelity modeling, Monte Carlo estimation, and reparameterization techniques. We validate it on nonlinear benchmarks and a real-world RANS-SST turbulence model calibration task. Results demonstrate unbiased EIG estimation with variance reduced by one to two orders of magnitude compared to single-fidelity methods, significantly enhancing the efficiency of Bayesian experimental design.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Economic aspects and information design of Web, GenAI, and cloud computing

Abstract

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.

Problem

• Estimating expected information gain for nonlinear Bayesian experimental design
• Reducing computational cost of high-fidelity model evaluations
• Developing unbiased multi-fidelity estimator with variance reduction

Innovation

• Multi-fidelity estimator reduces computational cost
• Unbiased variance minimization under budget constraints
• Reparameterization enables compatibility with control variates