Strictly proper kernel scores and characteristic kernels on compact spaces

TL;DR

This paper establishes the equivalence of strictly proper kernel scores and characteristic kernels, explores their limitations on infinite-dimensional spaces, and characterizes their existence on compact spaces, with applications in meteorology.

Contribution

It demonstrates the equivalence of proper kernel scores and characteristic kernels, characterizes their properties on compact spaces, and investigates their applicability to spherical domains.

Findings

Characteristic kernels cannot distinguish distributions far apart in total variation on infinite-dimensional spaces.

Existence of characteristic kernels on compact spaces is equivalent to the space being metrizable.

Special cases include translation-invariant kernels on compact Abelian groups and isotropic kernels on spheres.

Abstract

Strictly proper kernel scores are well-known tool in probabilistic forecasting, while characteristic kernels have been extensively investigated in the machine learning literature. 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. In addition, we provide a characterization of characteristic kernels in terms of eigenvalues and -functions 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.