  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.
