Sequential and adaptive probabilistic integration for Bayesian model updating

The present study attempts to introduce a Bayesian perspective to marginal likelihood estimation. For this purpose, the Bayesian marginal likelihood inference (BMLI) framework is first developed, where the posterior statistics of the marginal likelihood are estimated using efficient random process sampling. The sequential and adaptive probabilistic integration (SAPI) method is then proposed within the BMLI framework to efficiently infer the posterior distribution of the model parameters as a result of the marginal likelihood estimation. In this method, an adaptive tempering of the likelihood function is presented to draw samples from a series of intermediate distributions to perform probabilistic integration sequentially, and the samples are finally converged to the target posterior domain. An active learning process is also designed to accelerate the sequential integration with a minimum number of likelihood function observations. Four numerical examples are investigated and the superiority of the SAPI method over other sequential sampling methods is demonstrated.