K. Fujiwara and S. Watanabe (Japan)

Bayesian hypothesis testing, Bayesian marginal likelihood ratio, Bayesian factor, Gaussian Mixture Model, singular learning machine.

Statistical hypothesis testing is to compare the probability of an alternative hypothesis with a null hypothesis. The log likelihood ratio using the maximum likelihood estimator is widely used for regular model testing. However it is not appropriate for testing statistical singular models, as the maximum likelihood estimator diverges in singular mod els. Although Bayesian hypothesis testing by Bayesian log likelihood ratio is proved the most powerful test for the sta tistical singular models, most of problems have not been solved. In this paper, we proved the theoretical asymptotic distribution of the Bayesian log likelihood of a problem to determine the number of components of a Gaussian mix ture model by Bayesian hypothesis testing and algebraic geometrical method. We also show the experimental result how the Bayesian hypothesis testing works well.

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