Strictly proper kernel scores and characteristic kernels on compact spaces

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