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Stochastic Reasoning, Free Energy, and Information Geometry

shiro{at}ism.ac.jp, Institute of Statistical Mathematics, Tokyo 106-8569, Japan, and Gatsby Computational Neuroscience Unit, University College London, London WC1N 3AR, U.K.

Toshiyuki Tanaka

tanaka{at}eei.metro-u.ac.jp, Department of Electronics and Information Engineering, Tokyo Metropolitan University, Tokyo 192-0397, Japan

Shun-ichi Amari

amari{at}brain.riken.jp, RIKEN Brain Science Institute, Saitama 351-0198, Japan

Belief propagation (BP) is a universal method of stochastic reasoning. It gives exact inference for stochastic models with tree interactions and works surprisingly well even if the models have loopy interactions. Its performance has been analyzed separately in many fields, such as AI, statistical physics, information theory, and information geometry. This article gives a unified framework for understanding BP and related methods and summarizes the results obtained in many fields. In particular, BP and its variants, including tree reparameterization and concave-convex procedure, are reformulated with information-geometrical terms, and their relations to the free energy function are elucidated from an information-geometrical viewpoint. We then propose a family of new algorithms. The stabilities of the algorithms are analyzed, and methods to accelerate them are investigated.

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