In this paper, we study error bounds for {\em Bayesian quadrature} (BQ), with an emphasis on noisy settings, randomized algorithms, and average-case performance measures. We seek to approximate the integral of functions in a {\em Reproducing Kernel Hilbert Space} (RKHS), particularly focusing on the Mat\'ern-$\nu$ and squared exponential (SE) kernels, with samples from the function potentially being corrupted by Gaussian noise. We provide a two-step meta-algorithm that serves as a general tool for relating the average-case quadrature error with the $L^2$-function approximation error. When specialized to the Mat\'ern kernel, we recover an existing near-optimal error rate while avoiding the existing method of repeatedly sampling points. When specialized to other settings, we obtain new average-case results for settings including the SE kernel with noise and the Mat\'ern kernel with misspecification. Finally, we present algorithm-independent lower bounds that have greater generality and/or give distinct proofs compared to existing ones.

翻译：本文研究了贝叶斯求积（BQ）的误差界，重点关注含噪设置、随机化算法和平均情况性能指标。我们寻求近似再生核希尔伯特空间（RKHS）中函数的积分，特别关注Matérn-ν核和平方指数（SE）核，其中函数的样本可能被高斯噪声污染。我们提出了一种两步元算法，作为连接平均情况求积误差与L²函数逼近误差的通用工具。当专门应用于Matérn核时，我们恢复了现有接近最优的误差率，同时避免了重复采样点的现有方法。当应用于其他设置时，我们获得了新设置下的平均情况结果，包括含噪SE核和设定错误的Matérn核。最后，我们提出了与现有方法相比具有更大通用性和/或不同证明的算法无关下界。