Posterior consistency for nonparametric Hidden Markov Models with finite state space

In this paper we study posterior consistency for different topologies on the parameters for hidden Markov models with finite state space. We first obtain weak and strong posterior consistency for the marginal density function of finitely many consecutive observations. We deduce posterior consistency for the different components of the parameter. We finally apply our results to independent emission probabilities, translated emission probabilities and discrete HMMs, some priors for which the posterior is consistent are given.