Previously, we proposed a probabilistic data generation model represented by an unobservable tree and a sequential updating method to calculate a posterior distribution over a set of trees. The set is called a meta-tree. In this paper, we propose a more efficient batch updating method.

The test of homogeneity for normal mixtures has been used in various fields, but its theoretical understanding is limited because the parameter set for the null hypothesis corresponds to singular points in the parameter space. In this paper, we shed a light on this issue from a new perspective, variational Bayes, and offer a theory for testing homogeneity based on it. Conventional theory has not reveal the stochastic behavior of the variational free energy, which is necessary for constructing a hypothesis test, has remained unknown. We clarify it for the first time and construct a new test base on it. Numerical experiments show the validity of our results.

We analyze the extended stochastic complexity (ESC) which has been proposed by K. Yamanishi. The ESC can be applied to learning algorithms for on-line prediction and batch-learning settings. Yamanishi derived the upper bound of ESC satisfying uniformly for all data sequences and that of the asymptotic expectation of ESC. However, Yamanishi concentrates mainly on the worst case performance and the lower bound has not been derived. In this paper, we show some interesting properties of ESC which are similar to Bayesian statistics: the Bayes rule and the asymptotic normality. We then derive the asymptotic formula of ESC in the meaning of almost sure and mean convergence within an error of o(1) using these properties.