On the asymptotic behaviour of the posterior distribution in hidden Markov Models
Résumé
In this paper, we investigate the asymptotic behaviour of the posterior distribution in hidden Markov models (HMMs). We obtain a general asymptotic result, and give conditions on the prior under which we obtain a rate of convergence for the posterior distribution of the marginal distributions of the process. We then focus on the situation where the hidden Markov chain evolves on a finite state space but where the number of hidden states might be larger than the true one. It is known that the likelihood ratio test statistic for overfitted HMMs has a non standard behavior and is unbounded. Our conditions on the prior may be seen as a way to penalize parameters to avoid this phenomenon. We are then able to define a consistent Bayesian estimator of the number of hidden states. We also give a precise description of the situation when the observations are i.i.d. and we allow $2$ possible hidden states. Some simulations are presented to illustrate our results.
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