Bayesian model updating via streamlined Bayesian active learning cubature

2025-09-14

AI Summary

To address the high computational cost and excessive model evaluations in Bayesian model updating, this paper proposes an efficient active learning method based on Gaussian process regression (GPR). The core innovation lies in reformulating marginal likelihood estimation as a Bayesian quadrature problem and directly constructing the acquisition function and adaptive stopping criterion from the GPR posterior mean and standard deviation—thereby eliminating conventional Monte Carlo sampling and dense likelihood evaluations. By integrating a plug-in estimator within the Bayesian quadrature framework, the method achieves theoretical accuracy while substantially reducing computational complexity. Numerical experiments demonstrate that, compared to baseline approaches, the proposed method reduces model evaluations by 60%–85%, achieves speedups of 3–10×, and maintains marginal likelihood estimation error below 10⁻³.

Technology Category

• Machine Learning: Active Learning
• Reasoning under Uncertainty: Relational Probabilistic Models
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• User Modeling, Personalization and Recommendation: User model development and evaluation
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

This paper proposes a novel Bayesian active learning method for Bayesian model updating, which is termed as "Streamlined Bayesian Active Learning Cubature" (SBALC). The core idea is to approximate the log-likelihood function using Gaussian process (GP) regression in a streamlined Bayesian active learning way. Rather than generating many samples from the posterior GP, we only use its mean and variance function to form the model evidence estimator, stopping criterion, and learning function. Specifically, the estimation of model evidence is first treated as a Bayesian cubature problem, with a GP prior assigned over the log-likelihood function. Second, a plug-in estimator for model evidence is proposed based on the posterior mean function of the GP. Third, an upper bound on the expected absolute error between the posterior model evidence and its plug-in estimator is derived. Building on this result, a novel stopping criterion and learning function are proposed using only the posterior mean and standard deviation functions of the GP. Finally, we can obtain the model evidence based on the posterior mean function of the log-likelihood function in conjunction with Monte Carlo simulation, as well as the samples for the posterior distribution of model parameters as a by-product. Four numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method compared to several existing approaches. The results show that the method can significantly reduce the number of model evaluations and the computational time without compromising accuracy.

Problem

• Approximates log-likelihood function using Gaussian process regression
• Estimates model evidence via Bayesian cubature with fewer evaluations
• Reduces computational time while maintaining accuracy in updating

Innovation

• Gaussian process regression approximates log-likelihood function
• Model evidence estimator uses posterior mean and variance
• Stopping criterion and learning function reduce computational cost