  The use of non parametric hidden Markov models with finite state space is
flourishing in practice while few theoretical guarantees are known in this
framework. Here, we study asymptotic guarantees for these models in the
Bayesian framework. We obtain posterior concentration rates with respect to the
$L_1$-norm on joint marginal densities of consecutive observations in a general
theorem. We apply this theorem to two cases and obtain minimax concentration
rates. We consider discrete observations with emission distributions
distributed from a Dirichlet process and continuous observations with emission
distributions distributed from Dirichlet process mixtures of Gaussian
distributions.
