Algebraic geometry and stochastic complexity of hidden Markov models

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Hidden Markov models are now used in many fields, for example, speech recognition, natural language processing, etc. However, the mathematical foundation of analysis for the models is not yet constructed, since the HMM is non-identifiable. In recent years, we have developed the algebraic geometrical method that allows us to analyze the non-regular and non-identifiable models. In this paper, we apply this method to the HMM and reveal the asymptotic stochastic complexity in a mathematically rigorous way. Our results show that the Bayesian estimation makes the generalization error small and that the well known BIC is different from the stochastic complexity.