On the asymptotic behaviour of the posterior distribution in hidden Markov Models with unknown number of statesReport as inadecuate




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1 LM-Orsay - Laboratoire de Mathématiques d-Orsay 2 CREST - Centre de Recherche en Économie et Statistique 3 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

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 non standard 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.

Keywords : hidden Markov models Bayesian statistics posterior consistency posterior consistency.





Author: Elisabeth Gassiat - Judith Rousseau -

Source: https://hal.archives-ouvertes.fr/



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