  We consider the problem of estimation in Hidden Markov models with finite
state space and nonparametric emission distributions. Efficient estimators for
the transition matrix are exhibited, and a semiparametric Bernstein-von Mises
result is deduced. Following from this, we propose a modular approach using the
cut posterior to jointly estimate the transition matrix and the emission
densities. We derive a general theorem on contraction rates for this approach.
We then show how this result may be applied to obtain a contraction rate result
for the emission densities in our setting; a key intermediate step is an
inversion inequality relating $L^1$ distance between the marginal densities to
$L^1$ distance between the emissions. Finally, a contraction result for the
smoothing probabilities is shown, which avoids the common approach of sample
splitting. Simulations are provided which demonstrate both the theory and the
ease of its implementation.
