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Biometrika Advance Access originally published online on February 28, 2007
Biometrika 2007 94(1):19-35; doi:10.1093/biomet/asm018
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Copyright © 2007 Biometrika Trust

Articles

Model selection and estimation in the Gaussian graphical model

Ming Yuan

School of Industrial and Systems Engineering, Georgia Institute of Technology, 755 Ferst Drive NW, Atlanta, Georgia 30332, U.S.A.

Yi Lin

Department of Statistics, University of Wisconsin, Madison, Wisconsin 53706, U.S.A.

myuan{at}isye.gatech.edu

yilin{at}stat.wisc.edu

Received for publication 1 January 2006. Revision received 1 August 2006.
  Abstract

We propose penalized likelihood methods for estimating the concentration matrix in the Gaussian graphical model. The methods lead to a sparse and shrinkage estimator of the concentration matrix that is positive definite, and thus conduct model selection and estimation simultaneously. The implementation of the methods is nontrivial because of the positive definite constraint on the concentration matrix, but we show that the computation can be done effectively by taking advantage of the efficient maxdet algorithm developed in convex optimization. We propose a BIC-type criterion for the selection of the tuning parameter in the penalized likelihood methods. The connection between our methods and existing methods is illustrated. Simulations and real examples demonstrate the competitive performance of the new methods.

Key Words: covariance selection • lasso • maxdet algorithm • nonnegative garrote • penalized likelihood


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