Determine the Number of States in Hidden Markov Models via Marginal Likelihood
arxiv(2024)
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.
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