A Locally Adaptive Bayesian Cubature Method

Abstract

Bayesian cubature (BC) is a popular inferential perspective on the cubature of expensive integrands, wherein the integrand is emulated using a stochastic process model. Several approaches have been put forward to encode sequential adaptation (i.e. dependence on previous integrand evaluations) into this framework. However, these proposals have been limited to either estimating the parameters of a stationary covariance model or focusing computational resources on regions where large values are taken by the integrand. In contrast, many classical adaptive cubature methods focus computational resources on spatial regions in which local error estimates are largest. The contributions of this work are three-fold: First, we present a theoretical result that suggests there does not exist a direct Bayesian analogue of the classical adaptive trapezoidal method. Then we put forward a novel BC method that has empirically similar behaviour to the adaptive trapezoidal method. Finally we present evidence that the novel method provides improved cubature performance, relative to standard BC, in a detailed empirical assessment.