Goal-Oriented Bayesian Optimal Experimental Design for Nonlinear Models using Markov Chain Monte Carlo

2024-03-26

arXiv.org

AI Summary

Quantifying uncertainty in quantities of interest (QoIs) for nonlinear models requires experiment designs that directly target predictive uncertainty reduction, rather than conventional parameter-level uncertainty. Method: We propose a goal-oriented Bayesian optimal experimental design (OED) framework that maximizes the expected information gain (EIG) about the QoI—rather than model parameters—under nonlinear observation and prediction operators. We develop a theoretical foundation for QoI-oriented OED, introduce a nested Monte Carlo estimator combining MCMC-based posterior sampling with kernel density estimation for high-accuracy EIG computation, and employ Bayesian optimization to efficiently identify optimal experimental configurations. Results: Evaluated on multiple benchmark problems and a convection–diffusion source inversion task for sensor placement, our approach achieves significantly reduced QoI uncertainty and superior experimental efficiency compared to parameter-oriented OED, establishing a new paradigm for prediction-driven scientific experimentation.

Technology Category

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

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Optimal experimental design (OED) provides a systematic approach to quantify and maximize the value of experimental data. Under a Bayesian approach, conventional OED maximizes the expected information gain (EIG) on model parameters. However, we are often interested in not the parameters themselves, but predictive quantities of interest (QoIs) that depend on the parameters in a nonlinear manner. We present a computational framework of predictive goal-oriented OED (GO-OED) suitable for nonlinear observation and prediction models, which seeks the experimental design providing the greatest EIG on the QoIs. In particular, we propose a nested Monte Carlo estimator for the QoI EIG, featuring Markov chain Monte Carlo for posterior sampling and kernel density estimation for evaluating the posterior-predictive density and its Kullback-Leibler divergence from the prior-predictive. The GO-OED design is then found by maximizing the EIG over the design space using Bayesian optimization. We demonstrate the effectiveness of the overall nonlinear GO-OED method, and illustrate its differences versus conventional non-GO-OED, through various test problems and an application of sensor placement for source inversion in a convection-diffusion field.

Problem

• Nonlinear Models
• Predictive Target-oriented Optimization
• Information Gain Maximization

Innovation

• GO-OED
• Bayesian Optimization
• Nonlinear Model