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

• Published in: Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability Vol. 20; no. 4; pp. 2039 - 2075
• Main Authors: GASSIAT, ELISABETH, ROUSSEAU, JUDITH
• Format: Journal Article
• Language: English
• Published: International Statistical Institute and Bernoulli Society for Mathematical Statistics and Probability 01-11-2014; Bernoulli Society for Mathematical Statistics and Probability
• Subjects: Bayesian statistics; Consistent estimators; Density; Ergodic theory; hidden Markov models; Infinity; Integers; Markov chains; Markov models; Mathematical theorems; number of components; order selection; Polynomials; posterior distribution; Time series models
• ISSN: 1350-7265
• DOI: 10.3150/13-BEJ550

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.