Research on statistical inference of the graphical model by using graph theoretical and combinatorial commutative algebraic techniques

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This project focuses mainly on the following problems in the inference of the graphical model, (i) conditional exact tests of goodness-of-fit of the model with Markov bases (ii) localization of the algorithms of the inference (iii) Bayes estimation of parameters. We obtained the following results. (i) In many practical toric models we give explicit forms of Markov bases and proposed algorithms of conditional exact tests by using them. (ii) We proposed a localized algorithm of the iterative propotional scaling for computing the maximum likelihood estimator of a covariance matrix in Gauss graphical models. (iii) We proposed some Bayes admissible estimators of cell probabilities for Poisson decomposable graphical models.