Stochastic complexities of hidden Markov models

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 has not yet been constructed, since the HMMs are 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 order of its stochastic complexity in the mathematically rigorous way.