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

Elisabeth Gassiat; Judith Rousseau

doi:10.3150/13-bej550

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arXiv · 1207.2064

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

Elisabeth Gassiat·Judith Rousseau

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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.

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Elisabeth Gassiat, Judith Rousseau. 2014-10-24. About the posterior distribution in hidden Markov models with unknown number of states. https://doi.org/10.3150/13-bej550

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