Prequential probability: game-theoretic = measure theoretic

TL;DR

This paper demonstrates that in the prequential framework for binary outcome prediction, game-theoretic and measure-theoretic probabilities coincide on analytic sets, unifying two classical notions of probability.

Contribution

It proves the equivalence of game-theoretic and measure-theoretic probabilities in the prequential setting for binary outcomes, clarifying their relationship.

Findings

Game-theoretic and measure-theoretic probabilities coincide on analytic sets.

The result applies specifically to binary outcome prediction.

Unifies two classical probability notions in the prequential framework.

Abstract

This article continues study of the prequential framework for evaluating a probability 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 case of predicting binary outcomes, the two notions of probability in fact coincide on the analytic sets (in particular, on the Borel sets).