  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.
