A data driven equivariant approach to constrained Gaussian mixture modeling

Maximum likelihood estimation of Gaussian mixture models with differ-ent class-specific covariance matrices is known to be problematic. This is due to theunboundedness of the likelihood, together with the presence of spurious maximizers.Existing methods to bypass this obstacle are based on the fact that unboundedness isavoided if the eigenvalues of the covariance matrices are bounded away from zero. Thiscan be done imposing some constraints on the covariance matrices, i.e. by incorporat-ingaprioriinformation on the covariance structure of the mixture components. Thepresent work introduces a constrained approach, where the class conditional covari-ance matrices are shrunk towards a pre-specified target matrixΨ.Data-driven choicesof the matrixΨ,whenaprioriinformation is not available, and the optimal amountof shrinkage are investigated. Then, constraints based on a data-drivenΨare shown tobe equivariant with respect to linear affine transformations, provided that the methodused to select the target matrix be also equivariant. The effectiveness of the proposalis evaluated on the basis of a simulation study and an empirical example