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A non asymptotic penalized criterion for Gaussian mixture model selection

Published online by Cambridge University Press:  05 January 2012

Cathy Maugis  and
Bertrand Michel
Cathy Maugis
Affiliation:
Institut de Mathématiques de Toulouse, INSA de Toulouse, Université de Toulouse, 135 avenue de Rangueil, 31077 Toulouse Cedex 4, France; cathy.maugis@insa-toulouse.fr
Bertrand Michel
Affiliation:
Laboratoire de Statistique Théorique et Appliquée, Université Paris 6, 175 rue du Chevaleret, 75013 Paris, France; bertrand.michel@upmc.fr
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Abstract

Specific Gaussian mixtures are considered to solve simultaneouslyvariable selection and clustering problems. A non asymptoticpenalized criterion is proposed to choose the number of mixturecomponents and the relevant variable subset. Because of the nonlinearity of the associated Kullback-Leibler contrast on Gaussianmixtures, a general model selection theorem for maximum likelihoodestimation proposed by [Massart Concentration inequalities and model selection Springer, Berlin (2007). Lectures from the 33rd Summer School on Probability Theory held in Saint-Flour, July 6–23 (2003)] is used to obtainthe penalty function form. This theorem requires to control thebracketing entropy of Gaussian mixture families. The ordered andnon-ordered variable selection cases are both addressed in thispaper.

Keywords

Model-based clusteringvariable selectionpenalizedlikelihood criterionbracketing entropy
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 15: Supplement: In honor of Marc Yor , 2011 , pp. 41 - 68
Copyright
© EDP Sciences, SMAI, 2011

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