Hidden Markov models (HMM) have been widely used by scientists to model stochastic systems: the underlying process is a discrete Markov chain, and the observations are noisy realizations of the underlying process. Determining the number of hidden states for an HMM is a model selection problem which is yet to be satisfactorily solved, especially for the popular Gaussian HMM with heterogeneous covariance. In this paper, we propose a consistent method for determining the number of hidden states of HMM based on the marginal likelihood, which is obtained by integrating out both the parameters and hidden states. Moreover, we show that the model selection problem of HMM includes the order selection problem of finite mixture models as a special case. We give rigorous proof of the consistency of the proposed marginal likelihood method and provide an efficient computation method for practical implementation. We numerically compare the proposed method with the Bayesian information criterion (BIC), demonstrating the effectiveness of the proposed marginal likelihood method.

To characterize the Kullback-Leibler divergence and Fisher information in general parametrized hidden Markov models, in this paper, we first show that the log likelihood and its derivatives can be represented as an additive functional of a Markovian iterated function system, and then provide explicit characterizations of these two quantities through this representation. Moreover, we show that Kullback-Leibler divergence can be locally approximated by a quadratic function determined by the Fisher information. Results relating to the Cramér-Rao lower bound and the Hájek-Le Cam local asymptotic minimax theorem are also given. As an application of our results, we provide a theoretical justification of using Akaike information criterion (AIC) model selection in general hidden Markov models. Last, we study three concrete models: a Gaussian vector autoregressive-moving average model of order$(p,q)$, recurrent neural networks, and temporal restricted Boltzmann machine, to illustrate our theory.