Probability Theory for the Brier Game

Abstract

The usual theory of prediction with expert advice does not differentiate between good and bad “experts”: its typical results only assert that it is possible to efficiently merge not too extensive pools of experts, no matter how good or how bad they are. On the other hand, it is natural to expect that good experts' predictions will in some way agree with the actual outcomes (e.g., they will be accurate on the average). In this paper we show that, in the case of the Brier prediction game (also known as the quadratic-loss game), the predictions of a good (in some weak and natural sense) expert must satisfy the law of large numbers (both strong and weak) and the law of the iterated logarithm; we also show that two good experts' predictions must be in asymptotic agreement. Finally, we briefly discuss possible extensions of our results to more general games; the limit theorems for sequences of events in conventional probability theory correspond to the log-loss game.