Online EM Algorithm for Hidden Markov Models

TL;DR

This paper introduces an online EM algorithm for hidden Markov models that combines reparameterization with recursive smoothing, providing a new approach for real-time parameter estimation in time series data.

Contribution

It proposes a novel online EM algorithm that integrates complete-data sufficient statistics and recursive smoothing, with analysis of its convergence properties and practical evaluation.

Findings

Algorithm achieves estimation accuracy comparable to maximum likelihood for large samples.

Provides insights into the limiting behavior and convergence points of the recursive algorithm.

Demonstrates effectiveness through simulations on noisy Markov chain observations.

Abstract

Online (also called "recursive" or "adaptive") estimation of fixed model parameters in hidden Markov models is a topic of much interest in times series modelling. In this work, we propose an online parameter estimation algorithm that combines two key ideas. The first one, which is deeply rooted in the Expectation-Maximization (EM) methodology consists in reparameterizing the problem using complete-data sufficient statistics. The second ingredient consists in exploiting a purely recursive form of smoothing in HMMs based on an auxiliary recursion. Although the proposed online EM algorithm resembles a classical stochastic approximation (or Robbins-Monro) algorithm, it is sufficiently different to resist conventional analysis of convergence. We thus provide limited results which identify the potential limiting points of the recursion as well as the large-sample behavior of the quantities involved in the algorithm. The performance of the proposed algorithm is numerically evaluated through simulations in the case of a noisily observed Markov chain. In this case, the algorithm reaches estimation results that are comparable to that of the maximum likelihood estimator for large sample sizes.