Bayesian Model Inference using Bayesian Quadrature: the Art of Acquisition Functions and Beyond

Abstract: Estimating posteriors and the associated model evidences is a core issue of Bayesian model inference, and can be of great challenge given complex features of the posteriors such as multi-modalities of unequal importance, nonlinear dependencies and high sharpness. Bayesian Quadrature (BQ) has emerged as a competitive framework for tackling this challenge, as it provides flexible balance between computational cost and accuracy. The performance of a BQ scheme is fundamentally dictated by the acquisition function as it exclusively governs the generation of integration points. After reexamining one of the most advanced acquisition function from a prospective inference perspective and reformulating the quadrature rules for prediction, four new acquisition functions, inspired by distinct intuitions on expected rewards, are primarily developed, all of which are accompanied by elegant interpretations and highly efficient numerical estimators. Mathematically, these four acquisition functions measure, respectively, the prediction uncertainty of posterior, the contribution to prediction uncertainty of evidence, as well as the expected reduction of prediction uncertainties concerning posterior and evidence, and thus provide flexibility for highly effective design of integration points. These acquisition functions are further extended to the transitional BQ scheme, along with several specific refinements, to tackle the above-mentioned challenges with high efficiency and robustness. Effectiveness of the developments is ultimately demonstrated with extensive benchmark studies and application to an engineering example.