Brier score

The Brier score is a proper score function that measures the accuracy of probabilistic predictions. It is applicable to tasks in which predictions must assign probabilities to a set of mutually exclusive discrete outcomes. The set of possible outcomes can be either binary or categorical in nature, and the probabilities assigned to this set of outcomes must sum to one (where each individual probability is in the range of 0 to 1). It was proposed by Glenn W. Brier in 1950.

The Brier score can be thought of as either a measure of the "calibration" of a set of probabilistic predictions, or as a "cost function". More precisely, across all items ${\displaystyle i\in {1...N}}$ in a set N predictions, the Brier score measures the mean squared difference between:

Therefore, the lower the Brier score is for a set of predictions, the better the predictions are calibrated. Note that the Brier score, in its most common formulation, takes on a value between zero and one, since this is the largest possible difference between a predicted probability (which must be between zero and one) and the actual outcome (which can take on values of only 0 and 1). In the original (1950) formulation of the Brier score, the range is double, from zero to two.

The Brier score is appropriate for binary and categorical outcomes that can be structured as true or false, but is inappropriate for ordinal variables which can take on three or more values (this is because the Brier score assumes that all possible outcomes are equivalently "distant" from one another).

The most common formulation of the Brier score is

In which ${\displaystyle f_{t}}$ is the probability that was forecast, ${\displaystyle o_{t}}$ the actual outcome of the event at instance t (0 if it does not happen and 1 if it does happen) and N is the number of forecasting instances. In effect, it is the mean squared error of the forecast. This formulation is mostly used for binary events (for example "rain" or "no rain"). The above equation is a proper scoring rule only for binary events; if a multi-category forecast is to be evaluated, then the original definition given by Brier below should be used.

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