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The continuous ranked probability score (CRPS) is a proper scoring rule. It can be understand as a generalisation of the absolute error to full predictive distribution. It also represents a generalisation of the Brier Score to predictive distributions.
The CRPS is defined as follows:
The CRPS is usually written as a negatively oriented score, meaning that small values are better. The score is unbounded (can go to infinity), except for cases in which the forecast target itself has bounded support. For example, if the values that the forecast target can assume are limited between 0 and 100, then the CRPS can also only assume finite values. This becomes intuitive when thinking about the CRPS as a generalisation of the absolute error, which can only grow to infinity when the forecast target itself is not bounded to a certain range.
You can add LaTeX math by writing it in between the tags <math> and </math>: <math>\sigma</math>