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Geometric Ergodicity in Hidden Markov Models

François Le Gland 1, Laurent Mevel 1, 2

(1996)

We consider an hidden Markov model with multidimensional observations, and with misspecification, i.e. the assumed coefficients (transition probability matrix, and observation conditional densities) are possibly different from the true coefficients. Under mild assumptions on the coefficients of both the true and the assumed models, we prove that : (i)~the prediction filter, and its gradient w.r.t. some parameter in the model, forget almost surely their initial condition exponentially fast, and (ii) the extended Markov chain, whose components are : the unobserved Markov chain, the observation sequence, the prediction filter, and its gradient, is geometrically ergodic and has a unique invariant probability distribution.

1:	SIGMA2 (INRIA - IRISA)
	CNRS : UMR6074 – INRIA – Institut National des Sciences Appliquées (INSA) - Rennes – Université de Rennes 1
2:	Institut de Recherche Mathématique de Rennes (IRMAR)
	CNRS : UMR6625 – Université de Rennes 1 – École normale supérieure de Cachan - ENS Cachan – Institut National des Sciences Appliquées (INSA) : - RENNES – Université de Rennes II - Haute Bretagne

Domain

:

Computer Science/Other

Keywords: HMM / MISSPECIFIED MODEL / PREDICTION FILTER / EXPONENTIAL FORGETTING / GEOMETRIC ERGODICITY / PRODUCT OF RANDOM MATRICES / BIRKHOFF CONTRACTION COEFFICIENT

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Submitted on: Wednesday, 24 May 2006 13:34:20

Updated on: Wednesday, 27 December 2006 13:48:54