The test of homogeneity for normal mixtures has been conducted in diverse research areas, but constructing a theory of the test of homogeneity is challenging because the parameter set for the null hypothesis corresponds to singular points in the parameter space. In this paper, we examine this problem from a new perspective and offer a theory of hypothesis testing for homogeneity based on a variational Bayes framework. In the conventional theory, the constant order term of the free energy has remained unknown, however, we clarify its asymptotic behavior because it is necessary for constructing a hypothesis test. Numerical experiments shows the validity of our theoretical results.
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