Mathematical foundation of efficient algorithms for statistical inference

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As a new mathematical method for analyzing propagation algorithms on graphs, a novel Graph zeta function has been proposed for the analysis of the loopy belief propagation algorithm, and used for revealing stronger theoretical results than the known ones. Also, a method of loop expansion has been proposed for analyzing the approximation error of the loopy belief propagation.As a mathematical foundation of the expectation propagation, an infinite dimensional exponential family has been proposed based on positive definite kernels, and its estimation methods and theoretical properties have been studied.