High dimensional Sparse Gaussian Graphical Mixture Model

Abstract: This paper considers the problem of networks reconstruction from heterogeneous data using a Gaussian Graphical Mixture Model (GGMM). It is well known that parameter estimation in this context is challenging due to large numbers of variables coupled with the degeneracy of the likelihood. We propose as a solution a penalized maximum likelihood technique by imposing an $l_{1}$ penalty on the precision matrix. Our approach shrinks the parameters thereby resulting in better identifiability and variable selection. We use the Expectation Maximization (EM) algorithm which involves the graphical LASSO to estimate the mixing coefficients and the precision matrices. We show that under certain regularity conditions the Penalized Maximum Likelihood (PML) estimates are consistent. We demonstrate the performance of the PML estimator through simulations and we show the utility of our method for high dimensional data analysis in a genomic application.