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

Rousseau, Judith; Gassiat, Elisabeth (2014), About the posterior distribution in hidden Markov models with unknown number of states, Bernoulli, 20, 4, p. 2039-2075. http://dx.doi.org/10.3150/13-BEJ550

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HMM-rev2.pdf (474.2Kb)
Type
Article accepté pour publication ou publié
Date
2014
Journal name
Bernoulli
Volume
20
Number
4
Publisher
International Statistical Institute
Pages
2039-2075
Publication identifier
http://dx.doi.org/10.3150/13-BEJ550
Metadata
Show full item record
Author(s)
Rousseau, Judith
Laboratoire de Mathématiques d'Orsay [LM-Orsay]
Centre de Recherche en Économie et Statistique [CREST]
Gassiat, Elisabeth
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)
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.
Subjects / Keywords
idden Markov models; Bayesian statistics; posterior consistency; posterior consistency.
JEL
C11 - Bayesian Analysis: General

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