Model selection and estimation in the Gaussian graphical model

We propose penalized likelihood methods for estimating the concentrationmatrix in the Gaussian graphical model. The methods lead toa sparse and shrinkage estimator of the concentration matrixthat is positive definite, and thus conduct model selectionand estimation simultaneously. The implementation of the methodsis nontrivial because of the positive definite constraint onthe concentration matrix, but we show that the computation canbe done effectively by taking advantage of the efficient maxdetalgorithm developed in convex optimization. We propose a BIC-typecriterion for the selection of the tuning parameter in the penalizedlikelihood methods. The connection between our methods and existingmethods is illustrated. Simulations and real examples demonstratethe competitive performance of the new methods.