Certified Bayesian optimal experimental design: from Expected Information Gain to Signal-to-Noise Ratio

2026-09-26

AI Summary

This study addresses the computational intractability of expected information gain (EIG) under nonlinear models, where traditional nested Monte Carlo sampling fails to scale. To overcome this limitation, the authors leverage logarithmic Sobolev inequalities and Fisher information matrix approximations to derive signal-to-noise ratio-based upper and lower bounds on EIG. Specifically, two classes of bounds—conservative and incremental—are proposed, which recover exact solutions in linear-Gaussian settings. The primary contribution of this work is a scalable Bayesian experimental design framework that combines interpretability with rigorous theoretical guarantees. By substantially reducing computational overhead, the proposed approach effectively overcomes the computational bottlenecks inherent in nonlinear scenarios, enabling practical large-scale applications.

Abstract

Optimal experimental design is often formulated as the maximization of the Expected Information Gain (EIG), which measures the expected reduction in uncertainty about a parameter of interest after observing the data. For nonlinear forward models and nonGaussian priors, computing the EIG typically requires costly nested Monte Carlo estimators or sophisticated density-approximation techniques. In this work, we derive new upper and lower bounds on the EIG using dimensional logarithmic Sobolev inequalities. The proposed bounds are expressed in terms of signal-to-noise ratios (SNR), preserving the simple matrix-based structure of the EIG in the linear-Gaussian setting and depending solely on covariance matrices and Fisher information matrices. We introduce two families of bounds: conservative bounds, designed to retain the entire information content, and incremental bounds, that prioritize early information acquisition for sequential experimental design. When model gradients are available, we further derive computable approximations of the Fisher-information terms that require only a small number of gradient evaluations, yielding substantial computational savings. The tightness of these bounds is controlled by the nonlinearity of the forward model, and in the linear-Gaussian setting, these bounds recover the exact value of the EIG. By eliminating the nested-sampling bottleneck, the proposed framework offers a tractable and theoretically grounded alternative to direct EIG estimation for large-scale Bayesian experimental design, while retaining the interpretability of the SNR-based bounds.

Problem

• Bayesian optimal experimental design
• Expected Information Gain
• nested Monte Carlo
• nonlinear forward models
• computational bottleneck

Innovation

• Bayesian optimal experimental design
• Expected Information Gain
• logarithmic Sobolev inequalities
• Signal-to-Noise Ratio
• Fisher information matrix