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 also obtain posterior consistency for marginal smoothing distributions in the discrete case. We finally apply our results to independent emission probabilities, translated emission probabilities and discrete HMMs, under various types of priors.