Bayesian Quadrature

Abstract:Bayesian quadrature is a probabilistic, model-based approach to numerical integration, the estimation of intractable integrals, or expectations. Although Bayesian quadrature was popularised already in the 1980s, no systematic and comprehensive treatment has been published. The purpose of this survey is to fill this gap. We review the mathematical foundations of Bayesian quadrature from different points of view; present a systematic taxonomy for classifying different Bayesian quadrature methods along the three axes of modelling, inference, and sampling; collect general theoretical guarantees; and provide a controlled numerical study that explores and illustrates the effect of different choices along the axes of the taxonomy. We also provide a realistic assessment of practical challenges and limitations to application of Bayesian quadrature methods and include an up-to-date and nearly exhaustive bibliography that covers not only machine learning and statistics literature but all areas of mathematics and engineering in which Bayesian quadrature or equivalent methods have seen use.

Machine Learning (cs.LG); Numerical Analysis (math.NA); Machine Learning (stat.ML)

arXiv:2602.16218 [cs.LG]

(or arXiv:2602.16218v2 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2602.16218 Focus to learn more arXiv-issued DOI via DataCite

Maren Mahsereci and Toni Karvonen (2026), "Bayesian Quadrature", Foundations and Trends in Machine Learning: Vol. 19, No. 2-3, pp 121-240

https://doi.org/10.1108/FTMAL-06-2026-0132 Focus to learn more DOI(s) linking to related resources

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