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.

翻译：贝叶斯求积是一种基于概率模型的数值积分方法，用于估计难解积分或数学期望。尽管贝叶斯求积早在20世纪80年代就已得到推广，但迄今尚未有系统全面的论述发表。本综述旨在填补这一空白。我们从不同视角回顾贝叶斯求积的数学基础；提出沿建模、推断和采样三个维度对贝叶斯求积方法进行系统分类的框架；汇总通用理论保证；并通过受控数值研究探索和阐释分类框架中各维度不同选择的影响。我们还对贝叶斯求积方法在实际应用中的挑战与局限进行了客观评估，并提供了近乎完备的最新参考文献目录，涵盖机器学习与统计学文献，以及贝叶斯求积或等效方法所涉及的所有数学与工程领域。