Hidden Markov processes

TLDR

Hidden Markov processes model a finite‑state Markov chain observed through a memoryless channel, and recent extensions have incorporated continuous state spaces and general alphabets, with the literature reviewing related topics. The paper provides an overview of the statistical and information‑theoretic aspects of hidden Markov processes. The authors develop new algorithms for estimating state, parameters, and order of HMPs, and for universal coding, classification, and decoding of hidden Markov channels. Statistical properties, ergodic theorems for relative entropy densities, and consistency and asymptotic normality of maximum‑likelihood estimators were established for HMPs, with analogous results for switching autoregressive processes.

Abstract

An overview of statistical and information-theoretic aspects of hidden Markov processes (HMPs) is presented. An HMP is a discrete-time finite-state homogeneous Markov chain observed through a discrete-time memoryless invariant channel. In recent years, the work of Baum and Petrie (1966) on finite-state finite-alphabet HMPs was expanded to HMPs with finite as well as continuous state spaces and a general alphabet. In particular, statistical properties and ergodic theorems for relative entropy densities of HMPs were developed. Consistency and asymptotic normality of the maximum-likelihood (ML) parameter estimator were proved under some mild conditions. Similar results were established for switching autoregressive processes. These processes generalize HMPs. New algorithms were developed for estimating the state, parameter, and order of an HMP, for universal coding and classification of HMPs, and for universal decoding of hidden Markov channels. These and other related topics are reviewed.