Consistency of the maximum likelihood estimator for general hidden Markov models

TL;DR

This paper proves the strong consistency of the maximum likelihood estimator for a broad class of hidden Markov models, including nonlinear, linear Gaussian, and finite state models, using an innovative information-theoretic approach.

Contribution

It introduces a novel information-theoretic method for establishing MLE consistency in general hidden Markov models without explicit entropy rate representation.

Findings

MLE is strongly consistent under minimal assumptions

Consistency results extend to nonlinear, Gaussian, and finite state models

Provides a new concentration inequality for ergodic Markov chains

Abstract

Consider a parametrized family of general hidden Markov models, where both the observed and unobserved components take values in a complete separable metric space. We prove that the maximum likelihood estimator (MLE) of the parameter is strongly consistent under a rather minimal set of assumptions. As special cases of our main result, we obtain consistency in a large class of nonlinear state space models, as well as general results on linear Gaussian state space models and finite state models. A novel aspect of our approach is an information-theoretic technique for proving identifiability, which does not require an explicit representation for the relative entropy rate. Our method of proof could therefore form a foundation for the investigation of MLE consistency in more general dependent and non-Markovian time series. Also of independent interest is a general concentration inequality for $V$-uniformly ergodic Markov chains.