MATHEMATICAL ENGINEERING TECHNICAL REPORTS Capital process and optimality properties of Bay.pdf

MATHEMATICAL ENGINEERING TECHNICAL REPORTS Capital process and optimality properties of Bayesian Skeptic in the fair and biased coin games Masayuki KUMON, Akimichi TAKEMURA and Kei TAKEUCHI METR 2005–32 October 2005 DEPARTMENT OF MATHEMATICAL INFORMATICS GRADUATE SCHOOL OF INFORMATION SCIENCE AND TECHNOLOGY THE UNIVERSITY OF TOKYO BUNKYO-KU, TOKYO 113-8656, JAPAN WWW page: http://www.i.u-tokyo.ac.jp/mi/mi-e.htm The METR technical reports are published as a means to ensure timely dissemination of scholarly and technical work on a non-commercial basis. Copyright and all rights therein are maintained by the authors or by other copyright holders, notwithstanding that they have offered their works here electronically. It is understood that all persons copying this information will adhere to the terms and constraints invoked by each author’s copyright. These works may not be reposted without the explicit permission of the copyright holder. Capital process and optimality properties of Bayesian Skeptic in the fair and biased coin games Masayuki Kumon Risk Analysis Research Center Institute of Statistical Mathematics Akimichi Takemura Graduate School of Information Science and Technology University of Tokyo and Kei Takeuchi Faculty of International Studies Meiji Gakuin University October, 2005 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( √ 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