Hierarchical Bayesian Quadrature

Tim Weiland, Toni Karvonen, Philipp Hennig
Proceedings of the 42nd Conference on Uncertainty in Artificial Intelligence, PMLR 337:7273-7289, 2026.

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v337-weiland26a, title = {Hierarchical {Bayesian} Quadrature}, author = {Weiland, Tim and Karvonen, Toni and Hennig, Philipp}, booktitle = {Proceedings of the 42nd Conference on Uncertainty in Artificial Intelligence}, pages = {7273--7289}, year = {2026}, editor = {Perković, Emilija and Malinsky, Daniel}, volume = {337}, series = {Proceedings of Machine Learning Research}, month = {17--21 Aug}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v337/main/assets/weiland26a/weiland26a.pdf}, url = {https://proceedings.mlr.press/v337/weiland26a.html}, 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.} }
Endnote
%0 Conference Paper %T Hierarchical Bayesian Quadrature %A Tim Weiland %A Toni Karvonen %A Philipp Hennig %B Proceedings of the 42nd Conference on Uncertainty in Artificial Intelligence %C Proceedings of Machine Learning Research %D 2026 %E Emilija Perković %E Daniel Malinsky %F pmlr-v337-weiland26a %I PMLR %P 7273--7289 %U https://proceedings.mlr.press/v337/weiland26a.html %V 337 %X 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.
APA
Weiland, T., Karvonen, T. & Hennig, P.. (2026). Hierarchical Bayesian Quadrature. Proceedings of the 42nd Conference on Uncertainty in Artificial Intelligence, in Proceedings of Machine Learning Research 337:7273-7289 Available from https://proceedings.mlr.press/v337/weiland26a.html.

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