In recent years, proper scoring rules have emerged as a power ful general approach for estimating probability distributions. In addition to significantly ex panding the range of modeling techniques that can be applied in practice, this has also substantially broadened the conceptual understanding of estimation methods. Originally, proper scoring rules we re conceived in meteorology as summary statistics for describing the performance of probabilisti c forecasts ( Murphy and Winkler, 1984), but they also play an important role in economics as tools for bel ief elicitation ( Schotter and Trevino, 2014). A probabilistic forecast is a probability distribution ove r the space of the possible outcomes of the future event that is stated by the forecaster. The simple st and most popular case of probabilistic forecasts arises when the outcome is binary, so the probabilistic forecast reduces to issuing a predictive probability of success. Brier ( 1950) was the first to consider the problem of devising a scoring rule which could not be "played" by a dishonest fore casting agent. He introduced the quadratic scoring rule and showed that it incentivizes a for ecasting agent to state his most accurate probability estimate when faced with uncertainty.

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.

There are two seemingly unrelated problems in insurance pricing that we are going to tackle in this paper. First, an insurance pricing system should not have any systematic cross-financing between different price cohorts. Systematic cross-financing implicitly means that some parts of the portfolio are under-priced, and this is compensated by other parts of the portfolio that are over-priced. We can prevent systematic cross-financing between price cohorts by ensuring that the pricing system is auto-calibrated. We propose to apply isotonic recalibration which turns any regression function into an auto-calibrated pricing system.