Hierarchical Bayesian Quadrature

2026-07-12

AI Summary

This work addresses the model misspecification inherent in traditional Bayesian quadrature when applied to non-stationary integrands, which arises from the use of stationary covariance functions. To overcome this limitation, the authors propose a hierarchical Bayesian quadrature framework that adaptively partitions the integration domain into locally stationary subregions via a tree-based decomposition. Within each subregion, a local Gaussian process model is employed, and estimates are coherently combined through a hierarchical conditioning scheme that preserves cross-region correlations. The approach avoids Markov chain Monte Carlo sampling, dynamically allocates computational budget according to local function complexity, and controls tree growth using principled model selection criteria. Empirical evaluations demonstrate that the method significantly outperforms existing approaches on non-stationary integrands while maintaining competitive accuracy in stationary settings.

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.

Problem

• Bayesian Quadrature
• nonstationary integrands
• numerical integration
• Gaussian process
• model misspecification

Innovation

• Hierarchical Bayesian Quadrature
• nonstationary integrands
• adaptive partitioning
• Gaussian process conditioning
• model selection