The variational Bayes learning approximates the Bayes posterior
distribution with smaller computational costs. Its
theoretical property is now being clarified, however, the
asymptotic behavior of the variational free energies in general
cases are still unknown.
In this paper, we clarify the upper and lower bounds of asymptotic
variational free energy when hyperparameters in the Dirichlet
distribution are different from each other. We show that our result
contains the conventional researches as special cases and that
the variational Bayes learning has a phase transition with respect to the
hyperparameter control.