Prequential probability: game-theoretic = measure theoretic

Abstract

This article continues study of the prequential framework for evaluating aprobability forecaster. Testing the hypothesis that the sequence of forecasts issued by the forecaster is in agreement with the observed outcomes can be done using prequential notions of probability. It turns out that there are two natural notions of probability in the prequential framework: game-theoretic, whose idea goes back to von Mises and Ville, and measure-theoretic, whose idea goes back to Kolmogorov. The main result of this article is that, in the caseof predicting binary outcomes, the two notions of probability in fact coincide on the analytic sets (in particular, on the Borel sets).