# Hierarchical Bayesian Quadrature

arXiv: https://arxiv.org/abs/2607.10793 · HTML: https://mlsanity.com/paper/2607.10793/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 35 references that are in the map.
- First posted on arXiv 12 Jul 2026.

## Builds on (most cited first, 18 of 18)

- [Scalable Global Optimization via Local Bayesian Optimization](https://mlsanity.com/paper/1910.01739/) (2019)
- [Distributed Gaussian Processes](https://mlsanity.com/paper/1502.02843/) (2015)
- [Probabilistic Numerics and Uncertainty in Computations](https://mlsanity.com/paper/1506.01326/) (2015)
- [Physics-Informed Gaussian Process Regression Generalizes Linear PDE Solvers](https://mlsanity.com/paper/2212.12474/) (2022)
- [Non-Stationary Spectral Kernels](https://mlsanity.com/paper/1705.08736/) (2017)
- [Sampling for Inference in Probabilistic Models with Fast Bayesian Quadrature](https://mlsanity.com/paper/1411.0439/) (2014)
- [Probabilistic Integration: A Role in Statistical Computation?](https://mlsanity.com/paper/1512.00933/) (2015)
- [Convergence Guarantees for Adaptive Bayesian Quadrature Methods](https://mlsanity.com/paper/1905.10271/) (2019)
- [Frank-Wolfe Bayesian Quadrature: Probabilistic Integration with Theoretical Guarantees](https://mlsanity.com/paper/1506.02681/) (2015)
- [Fast Bayesian Inference with Batch Bayesian Quadrature via Kernel Recombination](https://mlsanity.com/paper/2206.04734/) (2022)
- [Nested Kriging predictions for datasets with large number of observations](https://mlsanity.com/paper/1607.05432/) (2016)
- [Convergence Analysis of Deterministic Kernel-Based Quadrature Rules in Misspecified Settings](https://mlsanity.com/paper/1709.00147/) (2017)
- [Patchwork Kriging for Large-scale Gaussian Process Regression](https://mlsanity.com/paper/1701.06655/) (2017)
- [A Dictionary of Closed-Form Kernel Mean Embeddings](https://mlsanity.com/paper/2504.18830/) (2025)
- [Multilevel Bayesian Quadrature](https://mlsanity.com/paper/2210.08329/) (2022)
- [Bayesian Probabilistic Numerical Integration with Tree-Based Models](https://mlsanity.com/paper/2006.05371/) (2020)
- [Easily Computed Marginal Likelihoods from Posterior Simulation Using the THAMES Estimator](https://mlsanity.com/paper/2305.08952/) (2023)
- [Bayesian Quadrature](https://mlsanity.com/paper/2602.16218/) (2026)

## 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.