Kernel Stein discrepancy (KSD) is a widely used kernel-based measure of discrepancy between probability measures. It is often employed in the scenario where a user has a collection of samples from a candidate probability measure and wishes to compare them against a specified target probability measure. KSD has been employed in a range of settings including goodness-of-fit testing, parametric inference, MCMC output assessment and generative modelling. However, so far the method has been restricted to finite-dimensional data. We provide the first analysis of KSD in the generality of data lying in a separable Hilbert space, for example functional data. The main result is a novel Fourier representation of KSD obtained by combining the theory of measure equations with kernel methods. This allows us to prove that KSD can separate measures and thus is valid to use in practice. Additionally, our results improve the interpretability of KSD by decoupling the effect of the kernel and Stein operator. We demonstrate the efficacy of the proposed methodology by performing goodness-of-fit tests for various Gaussian and non-Gaussian functional models in a number of synthetic data experiments.

翻译：核斯坦因散度（KSD）是一种广泛使用的基于核的概率测度差异度量。它通常用于用户拥有来自候选概率测度的样本集，并希望将其与指定的目标概率测度进行比较的场景。KSD已应用于一系列场景，包括拟合优度检验、参数推断、马尔可夫链蒙特卡洛输出评估和生成建模。然而，到目前为止，该方法仅限于有限维数据。我们首次在数据位于可分离希尔伯特空间（例如函数型数据）的一般性下对KSD进行了分析。主要结果是通过将测度方程理论与核方法相结合，得到了一种新颖的KSD傅里叶表示。这使我们能够证明KSD可以分离测度，因此在实际应用中有效。此外，我们的结果通过解耦核与斯坦因算子的效应，提高了KSD的可解释性。我们通过在多种合成数据实验中对各种高斯和非高斯函数模型进行拟合优度检验，展示了所提出方法的有效性。