Neural Computation
Algebraic geometrical methods for hierarchical learning machines
Neural Networks
Algebraic Analysis for Singular Statistical Estimation
ALT '99 Proceedings of the 10th International Conference on Algorithmic Learning Theory
Algebraic Analysis for Nonidentifiable Learning Machines
Neural Computation
Asymptotic model selection for naive Bayesian networks
UAI'02 Proceedings of the Eighteenth conference on Uncertainty in artificial intelligence
Stochastic Complexities of Gaussian Mixtures in Variational Bayesian Approximation
The Journal of Machine Learning Research
Stochastic complexity for mixture of exponential families in generalized variational Bayes
Theoretical Computer Science
Upper bound for variational free energy of Bayesian networks
Machine Learning
A model selection method based on bound of learning coefficient
ICANN'06 Proceedings of the 16th international conference on Artificial Neural Networks - Volume Part II
Stochastic complexity for mixture of exponential families in variational bayes
ALT'05 Proceedings of the 16th international conference on Algorithmic Learning Theory
Upper bounds for variational stochastic complexities of bayesian networks
IDEAL'06 Proceedings of the 7th international conference on Intelligent Data Engineering and Automated Learning
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Bayesian networks arc now used in enormous fields, for example. system diagnosis. data mining, clusterings etc. In spite of wide range of their applications, the statistical properties have not yet bcen clarified because the models are nonidentifiable and non-regular. In a Bayesian network. the set of parameters for a smaller model is an analytic set with singularities in the parameter space of a large model. Because of these singularities, the Fisher information matrices are not positive definite. In other words, the mathematical foundation for learning has not been constructed. In recent years, however, we have developed a method to analyze nonregular models by using algebraic geometry. This method revealed the relation between model's singularities and its statistical properties. In this paper, applying this method to Bayesian networks with latent variables, we clarify the orders of the stochastic complexities. Our result shows that their upper bound is smaller than thc dimension of the parameter space. This means that the Bayesian generalization error is also far smaller than that of a regular model, and that Schwarz's model selection criterion BIC needs to be improved for Bayesian networks.