  In this work we establish the posterior consistency for a parametrized family
of partially observed, fully dominated Markov models. As a main assumption, we
suppose that the prior distribution assigns positive probability to all
neighborhoods of the true parameter, for a distance induced by the expected
Kullback-Leibler divergence between the family members' Markov transition
densities. This assumption is easily checked in general. In addition, under
some additional, mild assumptions we show that the posterior consistency is
implied by the consistency of the maximum likelihood estimator. The latter has
recently been established also for models with non-compact state space. The
result is then extended to possibly non-compact parameter spaces and
non-stationary observations. Finally, we check our assumptions on examples
including the partially observed Gaussian linear model with correlated noise
and a widely used stochastic volatility model.
