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We consider finite state space stationary hidden Markov models (HMMs) in thesituation where the number of hidden states is unknown. We provide afrequentist asymptotic evaluation of Bayesian analysis methods. Our main resultgives posterior concentration rates for the marginal densities, that is for thedensity of a fixed number of consecutive observations. Using conditions on theprior, we are then able to define a consistent Bayesian estimator of the numberof hidden states. It is known that the likelihood ratio test statistic foroverfitted HMMs has a nonstandard behaviour and is unbounded. Our conditions onthe prior may be seen as a way to penalize parameters to avoid this phenomenon.Inference of parameters is a much more difficult task than inference ofmarginal densities, we still provide a precise description of the situationwhen the observations are i.i.d. and we allow for $2$ possible hidden states.

About the posterior distribution in hidden Markov models with unknown number of states