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Bayesian sparse graphical models and their mixtures
Author(s) -
Talluri Rajesh,
Baladandayuthapani Veerabhadran,
Mallick Bani K.
Publication year - 2014
Publication title -
stat
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.61
H-Index - 18
ISSN - 2049-1573
DOI - 10.1002/sta4.49
Subject(s) - graphical model , lasso (programming language) , computer science , estimator , prior probability , model selection , bayesian inference , positive definiteness , bayesian probability , gaussian , algorithm , context (archaeology) , matrix (chemical analysis) , covariance matrix , mathematics , artificial intelligence , positive definite matrix , statistics , paleontology , eigenvalues and eigenvectors , physics , materials science , quantum mechanics , biology , world wide web , composite material
We propose Bayesian methods for Gaussian graphical models that lead to sparse and adaptively shrunk estimators of the precision (inverse covariance) matrix. Our methods are based on lasso‐type regularization priors leading to parsimonious parameterization of the precision matrix, which is essential in several applications involving learning relationships among the variables. In this context, we introduce a novel type of selection prior that develops a sparse structure on the precision matrix by making most of the elements exactly zero, in addition to ensuring positive definiteness—thus conducting model selection and estimation simultaneously. More importantly, we extend these methods to analyse clustered data using finite mixtures of Gaussian graphical model and infinite mixtures of Gaussian graphical models. We discuss appropriate posterior simulation schemes to implement posterior inference in the proposed models, including the evaluation of normalizing constants that are functions of parameters of interest, which result from the restriction of positive definiteness on the correlation matrix. We evaluate the operating characteristics of our method via several simulations and demonstrate the application to real‐data examples in genomics. Copyright © 2014 John Wiley & Sons, Ltd