About the posterior distribution in hidden Markov models with unknown number of states

Abstract

We consider finite state space stationary hidden Markov models (HMMs) in the situation where the number of hidden states is unknown. We provide a frequentist asymptotic evaluation of Bayesian analysis methods. Our main result gives posterior concentration rates for the marginal densities, that is for the density of a fixed number of consecutive observations. Using conditions on the prior, we are then able to define a consistent Bayesian estimator of the number of hidden states. It is known that the likelihood ratio test statistic for overfitted HMMs has a nonstandard behaviour and is unbounded. Our conditions on the prior may be seen as a way to penalize parameters to avoid this phenomenon. Inference of parameters is a much more difficult task than inference of marginal densities, we still provide a precise description of the situation when the observations are i.i.d. and we allow for $2$ possible hidden states.

Published in at http://dx.doi.org/10.3150/13-BEJ550 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

References in corpus (3)

• Identifiability of parameters in latent structure models with many observed variables
• Convergence rates of posterior distributions for noniid observations
• The local geometry of finite mixtures

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• Overfitting hidden Markov models with an unknown number of states
• Bayesian Higher Order Hidden Markov Models
• Asymptotic Normality of the Posterior Distributions in a Class of Hidden Markov Models
• Hypotheses testing and posterior concentration rates for semi-Markov processes