Hierarchical Bayesian Quadrature
2026-07-12 • Machine Learning
Machine Learning
AI summaryⓘ
The authors address a problem in numerical integration where traditional methods assume the function being integrated behaves the same across its entire range, which is not true for some complex functions. They propose a new method that splits the integration area into smaller parts, applying specialized models to each part that better fit local behavior. These local results are then combined carefully to consider relationships between parts, improving accuracy without complex computations. Their approach works better than traditional methods on functions that change behavior locally, while performing just as well on simpler functions.
Numerical integrationBayesian QuadratureGaussian processStationary covarianceNonstationary behaviorDomain partitioningModel selectionHierarchical modelsEpidemiological modelingProbabilistic machine learning
Authors
Tim Weiland, Toni Karvonen, Philipp Hennig
Abstract
Numerical integration is a cornerstone of various scientific computing applications, such as engineering simulations and model evidence computations in probabilistic machine learning. Bayesian Quadrature uses Gaussian process surrogates that explicitly encode structural assumptions about the integrand to obtain integral estimates with quantified uncertainty. These surrogates are predominantly based on stationary covariance functions, which results in model misspecification for integrands exhibiting nonstationary behavior. We tackle this issue through an adaptively growing, tree-based partition of the integration domain into local stationary models. Our method recombines the local integral estimates through a hierarchy of GP conditioning that reintroduces cross-subdomain correlations, while model selection criteria control the tree growth to avoid unnecessary partitioning. The resulting algorithm is simple, requires no MCMC, and adapts its evaluation budget to local integrand complexity. On benchmark integration problems and a model evidence computation for an epidemiological model, Hierarchical Bayesian Quadrature achieves substantial gains over standard Bayesian Quadrature on nonstationary integrands while matching its performance on stationary ones.