Strictly proper kernel scores are well-known tools in probabilistic forecasting, while characteristic kernels have been extensively investigated in machine learning. We show that both notions coincide, so insights from one part of the literature can be used in the other. We show that the metric induced by a characteristic kernel cannot reliably distinguish between distributions that are far apart in total variation norm as soon as the underlying space of measures is infinite dimensional. We describe characteristic kernels in terms of eigenvalues and eigenfunctions and apply this characterization to continuous kernels on (locally) compact spaces. In the compact case, we show that characteristic kernels exist if and only if the space is metrizable. As special cases we investigate translation-invariant kernels on compact Abelian groups and isotropic kernels on spheres. The latter are of interest for forecast evaluation of probabilistic predictions on spherical domains as encountered in meteorology and climatology.

在概率预测中，严格正确的内核分数是众所周知的工具，而在机器学习中已经广泛研究了特征内核。我们证明这两个概念是重合的，因此可以将一部分文献的见解用于另一部分。我们表明，只要度量的基础空间为无限维，特征核所引发的度量就无法可靠地区分总变异范数相距甚远的分布。我们根据特征值和特征函数描述特征核，并将此特征应用于（局部）紧空间上的连续核。在紧凑的情况下，我们表明，当且仅当空间可量化时，存在特征核。作为特殊情况，我们研究紧致Abelian群上的平移不变核和球面上的各向同性核。后者对于在气象和气候学中遇到的关于球形域的概率预测的预测评估很有用。