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.
