Bayesian model updating via streamlined Bayesian active learning cubature

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.