Scores for Multivariate Distributions and Level Sets

Evaluating Forecasts of Multivariate Probability Distributions Forecasts of multivariate probability distributions are required for a variety of applications. The availability of a score for a forecast is important for evaluating prediction accuracy, as well as estimating model parameters. In “Scores for Multivariate Distributions and Level Sets,” X. Meng, J. W. Taylor, S. Ben Taieb, and S. Li propose a theoretical framework that encompasses several existing scores for multivariate distributions and can be used to generate new scores. In some multivariate contexts, a forecast of a level set is needed, such as a density level set for anomaly detection or the level set of the cumulative distribution, which can be used as a measure of risk. This motivates consideration of scores for level sets. The authors show that such scores can be obtained by decomposing the scores developed for multivariate distributions. A simple numerical algorithm is presented to compute the scores, and practical applications are provided in the contexts of conditional value-at-risk for financial data and the combination of expert macroeconomic forecasts.