## Bernoulli

• Bernoulli
• Volume 20, Number 4 (2014), 2039-2075.

### 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.

#### Article information

Source
Bernoulli, Volume 20, Number 4 (2014), 2039-2075.

Dates
First available in Project Euclid: 19 September 2014

https://projecteuclid.org/euclid.bj/1411134453

Digital Object Identifier
doi:10.3150/13-BEJ550

Mathematical Reviews number (MathSciNet)
MR3263098

Zentralblatt MATH identifier
1302.62183

#### Citation

Gassiat, Elisabeth; Rousseau, Judith. About the posterior distribution in hidden Markov models with unknown number of states. Bernoulli 20 (2014), no. 4, 2039--2075. doi:10.3150/13-BEJ550. https://projecteuclid.org/euclid.bj/1411134453

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