# Strictly proper kernel scores and characteristic kernels on compact spaces

arXiv: https://arxiv.org/abs/1712.05279 · HTML: https://mlsanity.com/paper/1712.05279/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Dec 2017.

## Built on by (most cited first, 14 of 14)

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- [A Bias-Variance-Covariance Decomposition of Kernel Scores for Generative Models](https://mlsanity.com/paper/2310.05833/) (2023)
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- [Smoothed Distance Kernels for MMDs and Applications in Wasserstein Gradient Flows](https://mlsanity.com/paper/2504.07820/) (2025)
- [Signature Kernel Scoring Rule: A Spatio-Temporal Diagnostic for Probabilistic Weather Forecasting](https://mlsanity.com/paper/2510.19110/) (2025)
- [Uncertainty Quantification for Regression: A Unified Framework based on kernel scores](https://mlsanity.com/paper/2510.25599/) (2025)
- [Minimax Lower Bounds of Kernel Discrepancy Estimation: MMD, HSIC, KSD](https://mlsanity.com/paper/2607.24235/) (2026)
- [Efficient pooling of predictions via kernel embeddings](https://mlsanity.com/paper/2411.16246/) (2024)

## Abstract (arXiv)

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.