No. 963

Title:

Capital process and optimality properties of Bayesian Skeptic in the fair and biased coin games

Author(s):

Masayuki, Kumon (Risk Analysis Research Center, Institute of Statistical Mathematics);
Akimichi, Takemura (Graduate School of Information Science and Technology, University of Tokyo);
Kei, Takeuchi (Faculty of International Studies, Meiji Gakuin University)

Key words:

Azuma-Hoeffding-Bennet inequality, beta-binomial distribution, exchangeability, game-theoretic probability, hypergeometric distribution, Kullback divergence, prior distribution.

Abstract:

    We study capital process behavior in the fair-coin game and biased-coin games in the framework of the game-theoretic probability of Shafer and Vovk (2001). We show that if Skeptic uses a Bayesian strategy with a beta prior, the capital process is lucidly expressed in terms of the past average of Reality's moves. From this it is proved that the Skeptic's Bayesian strategy weakly forces the strong law of large numbers (SLLN) with the convergence rate of $O(\sqrt{\log n/n})$ and if Reality violates SLLN then the exponential growth rate of the capital process is very accurately described in terms of the Kullback divergence between the average of Reality's moves when she violates SLLN and the average when she observes SLLN. We also investigate optimality properties associated with Bayesian strategy.