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

TL;DR

This paper provides a comprehensive set of necessary and sufficient conditions ensuring the uniqueness of the invariant measure in the filtering process of ergodic hidden Markov models, solving a longstanding problem posed by Blackwell in 1957.

Contribution

It offers a complete characterization of Blackwell's ergodicity problem for hidden Markov chains, extending previous partial results and utilizing nonlinear filter stability theory.

Findings

Established necessary and sufficient conditions for invariant measure uniqueness.

Unified and extended earlier partial results on Blackwell's problem.

Applied nonlinear filter stability theory to solve the ergodicity problem.

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.