On Reducing Probability Forecasts to Yes/No Forecasts

Abstract

Some scores for yes/no forecasts discussed in a paper by Woodcock (1976) are further considered. An expression for a probability p0 is found for any score such that the expected value of the score is maximized if the event is forecast when its probability is greater than p0 and not forecast if its probability is less than p0. Particular expressions for p0 for the scores discussed by Woodcock are presented. Asymptotic values are mostly near either 0.5 or the sample relative frequency of the event. Comments are made on the dependence of p0 on sample size and order of verification. The relation of this note to earlier work by Bryan and Enger and also Thompson and Brier is briefly discussed.