Determine the Number of States in Hidden Markov Models via Marginal Likelihood

TL;DR

This paper introduces a consistent method based on marginal likelihood for determining the number of hidden states in Hidden Markov Models, addressing a key model selection challenge especially for Gaussian HMMs with complex covariance structures.

Contribution

It proposes a novel, theoretically justified marginal likelihood approach for HMM order selection, including a practical computation method and comparison with BIC.

Findings

The method is proven to be consistent.

Numerical experiments show its effectiveness over BIC.

Abstract

Hidden Markov models (HMM) have been widely used by scientists to model stochastic systems: the underlying process is a discrete Markov chain and the observations are noisy realizations of the underlying process. Determining the number of hidden states for an HMM is a model selection problem, which is yet to be satisfactorily solved, especially for the popular Gaussian HMM with heterogeneous covariance. In this paper, we propose a consistent method for determining the number of hidden states of HMM based on the marginal likelihood, which is obtained by integrating out both the parameters and hidden states. Moreover, we show that the model selection problem of HMM includes the order selection problem of finite mixture models as a special case. We give rigorous proof of the consistency of the proposed marginal likelihood method and provide an efficient computation method for practical implementation. We numerically compare the proposed method with the Bayesian information criterion (BIC), demonstrating the effectiveness of the proposed marginal likelihood method.