用高斯过程做积分，让数值积分更可靠且可量化误差。
Bayesian Quadrature: Gaussian Processes for Integration
- 用高斯过程建模被积函数，把积分问题转为概率推断。
- 系统梳理了方法分类与理论保证，验证不同设计的影响。
- 适合需要可信积分结果的科研人员，尤其关注不确定性分析者。
贝叶斯积分是一种基于模型的概率化数值积分方法，用于估计难以解析求解的积分或期望值。尽管该方法早在1980年代就已流行，但至今缺乏系统全面的综述。本文旨在填补这一空白：从多个角度回顾贝叶斯积分的数学基础；构建一个涵盖建模、推断和采样三个维度的方法分类体系；总结通用理论保证；并开展受控数值实验，展示分类轴上不同选择的实际影响。同时，对方法在实际应用中的挑战与局限给出现实评估，并提供一份覆盖机器学习、统计学及数学与工程各领域的最新、近乎完整的参考文献列表。
原文摘要 · Abstract (English)
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.
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