A Locally Adaptive Bayesian Cubature Method

TL;DR

This paper introduces a new Bayesian cubature method that adaptively allocates computational resources based on local error estimates, improving performance over standard Bayesian approaches.

Contribution

It presents a novel adaptive Bayesian cubature method inspired by classical adaptive trapezoidal rules, with theoretical insights and empirical validation.

Findings

The new method behaves similarly to the adaptive trapezoidal rule.

It demonstrates improved accuracy in empirical tests.

Theoretical analysis shows no direct Bayesian analogue of classical adaptive methods.

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.