A complete solution to Blackwell's unique ergodicity problem for hidden Markov chains

Abstract

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.

Published in at http://dx.doi.org/10.1214/10-AAP688 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (5)

• Uniform observability of hidden Markov models and filter stability for unstable signals
• Observability and nonlinear filtering
• The stability of conditional Markov processes and Markov chains in random environments
• A simple proof of Kaijser's unique ergodicity result for hidden Markov $\alpha$-chains
• On Markov chains induced by partitioned transition probability matrices

Cited by in corpus (6)

• On the Viterbi process with continuous state space
• On the exchange of intersection and supremum of sigma-fields in filtering theory
• Conditional ergodicity in infinite dimension
• Phase Transitions in Nonlinear Filtering
• Hidden Markov Models and the Bayes Filter in Categorical Probability
• Ergodicity, Decisions, and Partial Information