Scalar and Vector Partitions of the Probability Score: Part I. Two-State Situation

Abstract

Scalar and vector partitions of the probability score (PS) in the two-state situation are described and compared. These partitions, which are based upon expressions for the PS in which probability forecasts are considered to be scalars and vectors, respectively, provide similar, but not equivalent (i.e., linearly related), measures of the reliability and resolution of the forecasts. Specifically, the reliability (resolution) of the forecasts according to the scalar partition is, in general, greater (less) than their reliability (resolution) according to the vector partition. A sample collection of forecasts is used to illustrate the differences between these partitions. Several questions related to the use of scalar and vector partitions of the PS in the two-state situation are discussed, including the interpretation of the results of previous forecast evaluation studies and the relative merits of these partitions. The discussions indicate that the partition most often used in such studies has been a special ``scalar'' partition, a partition which is equivalent to the vector partition in the two-state situation, and that the vector partition is more appropriate than the scalar partition.