Johanna F. Ziegel

Mario V. Wüthrich, Johanna Ziegel

Insurance pricing systems should fulfill the auto-calibration property to ensure that there is no systematic cross-financing between different price cohorts. Often, regression models are not auto-calibrated. We propose to apply isotonic recalibration to a given regression model to ensure auto-calibration. Our main result proves that under a low signal-to-noise ratio, this isotonic recalibration step leads to explainable pricing systems because the resulting isotonically recalibrated regression functions have a low complexity.

Johanna Ziegel, David Ginsbourger, Lutz Dümbgen

We present new classes of positive definite kernels on non-standard spaces that are integrally strictly positive definite or characteristic. In particular, we discuss radial kernels on separable Hilbert spaces, and introduce broad classes of kernels on Banach spaces and on metric spaces of strong negative type. The general results are used to give explicit classes of kernels on separable $L^p$ spaces and on sets of measures.

Sam Allen, David Ginsbourger, Johanna Ziegel

Probabilistic predictions are probability distributions over the set of possible outcomes. Such predictions quantify the uncertainty in the outcome, making them essential for effective decision making. By combining multiple predictions, the information sources used to generate the predictions are pooled, often resulting in a more informative forecast. Probabilistic predictions are typically combined by linearly pooling the individual predictive distributions; this encompasses several ensemble learning techniques, for example. The weights assigned to each prediction can be estimated based on their past performance, allowing more accurate predictions to receive a higher weight. This can be achieved by finding the weights that optimise a proper scoring rule over some training data. By embedding predictions into a Reproducing Kernel Hilbert Space (RKHS), we illustrate that estimating the linear pool weights that optimise kernel-based scoring rules is a convex quadratic optimisation problem. This permits an efficient implementation of the linear pool when optimally combining predictions on arbitrary outcome domains. This result also holds for other combination strategies, and we additionally study a flexible generalisation of the linear pool that overcomes some of its theoretical limitations, whilst allowing an efficient implementation within the RKHS framework. These approaches are compared in an application to operational wind speed forecasts, where this generalisation is found to offer substantial improvements upon the traditional linear pool.

Ingo Steinwart, Johanna F. Ziegel

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.