title: The Bayesian Covariance Lasso.
abstract: Estimation of sparse covariance matrices and their inverse subject to positive definiteness constraints has drawn a lot of attention in recent years. The abundance of high-dimensional data, where the sample size (n) is less than the dimension (d), requires shrinkage estimation methods since the maximum likelihood estimator is not positive definite in this case. Furthermore, whennis larger thandbut not sufficiently larger, shrinkage estimation is more stable than maximum likelihood as it reduces the condition number of the precision matrix. Frequentist methods have utilized penalized likelihood methods, whereas Bayesian approaches rely on matrix decompositions or Wishart priors for shrinkage. In this paper we propose a new method, called the Bayesian Covariance Lasso (BCLASSO), for the shrinkage estimation of a precision (covariance) matrix. We consider a class of priors for the precision matrix that leads to the popular frequentist penalties as special cases, develop a Bayes estimator for the precision matrix, and propose an efficient sampling scheme that does not precalculate boundaries for positive definiteness. The proposed method is permutation invariant and performs shrinkage and estimation simultaneously for non-full rank data. Simulations show that the proposed BCLASSO performs similarly as frequentist methods for non-full rank data.
authors.affiliation: Department of Biostatistics University of North Carolina Chapel Hill, North Carolina 27599, url:
authors.forename: Zakaria S
authors.initials: ZS
authors.lastname: Khondker
authors.affiliation: Department of Biostatistics University of North Carolina Chapel Hill, North Carolina 27599.
authors.forename: Hongtu
authors.initials: H
authors.lastname: Zhu
authors.affiliation: Division of Biostatistics University of Minnesota Minneapolis, Minnesota, U.S.A.
authors.forename: Haitao
authors.initials: H
authors.lastname: Chu
authors.affiliation: Biomedical Research Imaging Center University of North Carolina Chapel Hill, North Carolina 27599.
authors.forename: Weili
authors.initials: W
authors.lastname: Lin
authors.affiliation: Department of Biostatistics University of North Carolina Chapel Hill, North Carolina 27599.
authors.forename: Joseph G
authors.initials: JG
authors.lastname: Ibrahim
keywords: Bayesian covariance lasso; Network exploration; Penalized likelihood; Precision matrix; non-full rank data