A measure of the accuracy of a set of probability assessments. Proposed by Brier (1950), it is the average deviation between predicted probabilities for a set of events and their outcomes, so a lower score represents higher accuracy. In practice, the Brier score is often calculated according to Murphy’s (1972) partition into three additive components. Murphy’s partition is applied to a set of probability assessments for independent-event forecasts when a single probability is assigned to each event:

where c is the overall proportion correct, c_t is the proportion correct in category t, p_t is the probability assessed for category t, n_t is the number of assessments in category t, and N is the total number of assessments. The first term reflects the base rate of the phenomenon for which probabilities are assessed (e.g., overall proportion of correct forecasts), the second is a measure of the calibration of the probability assessments, and the third is a measure of the resolution.  Lichtenstein, Fischhoff and Phillips (1982) provide a more complete discussion of the Brier score for the evaluation of probability assessments.

• Brier, G. W. (1950), “Verification of forecasts expressed in terms of probability,” Monthly Weather Review, 75, 1-3.
• Lichtenstein, S., B. Fischhoff & L. Phillips (1982), “Calibration of probabilities: The state of the art to 1980,” in Kahneman, D., P. Slovic & A. Tversky (eds), Judgment Under Uncertainty: Heuristics and Biases.  New York: Cambridge University Press.
• Murphy, A. H. (1972), “Scalar and vector partitions of the probability score (Part I), Two state situation,” Journal of Applied Meteorology, 11, 273-282.