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Hierarchical Bayesian Quadrature

arXiv 2607.10793 ↗First posted 12 Jul 2026cs.LG cs.NA math.NAcited by 0

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Abstract (arXiv)

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.

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References · 18

  1. 2019Scalable Global Optimization via Local Bayesian Optimizationcited by 374
  2. 2015Distributed Gaussian Processescited by 84
  3. 2015Probabilistic Numerics and Uncertainty in Computationscited by 60
  4. 2022Physics-Informed Gaussian Process Regression Generalizes Linear PDE Solverscited by 41
  5. 2017Non-Stationary Spectral Kernelscited by 38
  6. 2014Sampling for Inference in Probabilistic Models with Fast Bayesian Quadraturecited by 31
  7. 2015Probabilistic Integration: A Role in Statistical Computation?cited by 26
  8. 2019Convergence Guarantees for Adaptive Bayesian Quadrature Methodscited by 22
  9. 2015Frank-Wolfe Bayesian Quadrature: Probabilistic Integration with Theoretical Guaranteescited by 20
  10. 2022Fast Bayesian Inference with Batch Bayesian Quadrature via Kernel Recombinationcited by 20
  11. 2016Nested Kriging predictions for datasets with large number of observationscited by 18
  12. 2017Convergence Analysis of Deterministic Kernel-Based Quadrature Rules in Misspecified Settingscited by 16
6 more
  1. 2017Patchwork Kriging for Large-scale Gaussian Process Regressioncited by 9
  2. 2025A Dictionary of Closed-Form Kernel Mean Embeddingscited by 9
  3. 2022Multilevel Bayesian Quadraturecited by 8
  4. 2020Bayesian Probabilistic Numerical Integration with Tree-Based Modelscited by 5
  5. 2023Easily Computed Marginal Likelihoods from Posterior Simulation Using the THAMES Estimatorcited by 1
  6. 2026Bayesian Quadraturecited by 1

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