  We develop necessary and sufficient conditions for uniqueness of the
invariant measure of the filtering process associated to an ergodic hidden
Markov model in a finite or countable state space. These results provide a
complete solution to a problem posed by Blackwell (1957), and subsume earlier
partial results due to Kaijser, Kochman and Reeds. The proofs of our main
results are based on the stability theory of nonlinear filters.
