  Bayesian networks are now being used in enormous fields, for example,
diagnosis of a system, data mining, clustering and so on. In spite of their
wide range of applications, the statistical properties have not yet been
clarified, because the models are nonidentifiable and non-regular. In a
Bayesian network, the set of its parameter for a smaller model is an analytic
set with singularities in the space of large ones. Because of these
singularities, the Fisher information matrices are not positive definite. In
other words, the mathematical foundation for learning was not constructed. In
recent years, however, we have developed a method to analyze non-regular models
using algebraic geometry. This method revealed the relation between the models
singularities and its statistical properties. In this paper, applying this
method to Bayesian networks with latent variables, we clarify the order of the
stochastic complexities.Our result claims that the upper bound of those is
smaller than the dimension of the parameter space. This means that the Bayesian
generalization error is also far smaller than that of regular model, and that
Schwarzs model selection criterion BIC needs to be improved for Bayesian
networks.
