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Selecting Hidden Markov Model State Number with Cross-Validated Likelihood

Gilles Celeux 1 Jean-Baptiste Durand 2, *
* Corresponding author
1 SELECT - Model selection in statistical learning
LMO - Laboratoire de Mathématiques d'Orsay, Inria Saclay - Ile de France
2 MISTIS [2007-2015] - Modelling and Inference of Complex and Structured Stochastic Systems [2007-2015]
Grenoble INP [2007-2019] - Institut polytechnique de Grenoble - Grenoble Institute of Technology [2007-2019], LJK [2007-2015] - Laboratoire Jean Kuntzmann [2007-2015], Inria Grenoble - Rhône-Alpes
Abstract : The problem of estimating the number of hidden states in a hidden Markov model is considered. Emphasis is placed on cross-validated likelihood criteria. Using cross-validation to assess the number of hidden states allows to circumvent the well-documented technical difficulties of the order identification problem in mixture models. Moreover, in a predictive perspective, it does not require that the sampling distribution belongs to one of the models in competition. However, computing cross-validated likelihood for hidden Markov models for which only one training sample is available, involves difficulties since the data are not independent. Two approaches are proposed to compute cross-validated likelihood for a hidden Markov model. The first one consists of using a deterministic half-sampling procedure, and the second one consists of an adaptation of the EM algorithm for hidden Markov models, to take into account randomly missing values induced by cross-validation. Numerical experiments on both simulated and real data sets compare different versions of cross-validated likelihood criterion and penalised likelihood criteria, including BIC and a penalised marginal likelihood criterion. Those numerical experiments highlight a promising behaviour of the deterministic half-sampling criterion.
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Gilles Celeux, Jean-Baptiste Durand. Selecting Hidden Markov Model State Number with Cross-Validated Likelihood. Computational Statistics, Springer Verlag, 2008, 23 (4), pp.541-564. ⟨10.1007/s00180-007-0097-1⟩. ⟨inria-00193098⟩

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