Maximum mean discrepancies (MMDs) like the kernel Stein discrepancy (KSD) have grown central to a wide range of applications, including hypothesis testing, sampler selection, distribution approximation, and variational inference. In each setting, these kernel-based discrepancy measures are required to (i) separate a target P from other probability measures or even (ii) control weak convergence to P. In this article we derive new sufficient and necessary conditions to ensure (i) and (ii). For MMDs on separable metric spaces, we characterize those kernels that separate Bochner embeddable measures and introduce simple conditions for separating all measures with unbounded kernels and for controlling convergence with bounded kernels. We use these results on $\mathbb{R}^d$ to substantially broaden the known conditions for KSD separation and convergence control and to develop the first KSDs known to exactly metrize weak convergence to P. Along the way, we highlight the implications of our results for hypothesis testing, measuring and improving sample quality, and sampling with Stein variational gradient descent.

翻译：最大均值差异（MMD），如核斯坦因差异（KSD），在假设检验、采样器选择、分布近似和变分推断等广泛应用中日益核心。在每种场景下，这些基于核的差异度量需要（i）将目标分布P与其他概率测度分离，甚至（ii）控制弱收敛到P。本文推导了确保（i）和（ii）的充分必要条件。对于可分度量空间上的MMD，我们刻画了能够分离Bochner可嵌入测度的核，并引入了采用无界核分离所有测度以及采用有界核控制收敛的简单条件。我们将这些结果应用于$\mathbb{R}^d$，显著拓展了KSD分离与收敛控制的已知条件，并开发了首个能精确度量弱收敛到P的KSD。在此过程中，我们重点阐述了这些结果对假设检验、样本质量评估与改进，以及基于Stein变分梯度下降采样等领域的影响。