Generalization of the Ignorance Score: Continuous Ranked Version and Its Decomposition

The Brier score (BS) and its generalizations to the multicategory ranked probability score (RPS) and to the continuous ranked probability score (CRPS) are the prominent verification measures for probabilistic forecasts. Particularly, their decompositions into measures quantifying the reliability, resolution, and uncertainty of the forecasts are attractive. Information theory sets up the natural framework for forecast verification. Recently, it has been shown that the BS is a second-order approximation of the information-based ignorance score (IGN), which also contains easily interpretable components and can also be generalized to a ranked version (RIGN). Here, the IGN, its generalizations, and decompositions are systematically discussed in analogy to the variants of the BS. Additionally, a continuous ranked IGN (CRIGN) is introduced in analogy to the CRPS. The applicability and usefulness of the conceptually appealing CRIGN are illustrated, together with an algorithm to evaluate its components reliability, resolution, and uncertainty for ensemble-generated forecasts.