Posterior Convergence Rates for Estimating Large Precision Matrices Using Graphical Models

• Activity Number: 345
• Type: Topic Contributed
• Date/Time: Tuesday, August 6, 2013 : 10:30 AM to 12:20 PM
• Sponsor: Section on Bayesian Statistical Science
• Author(s): Sayantan Banerjee*+ and Subhashis Ghosal
• Companies: North Carolina State University and North Carolina State University
• Keywords: Precision matrix ; G-Wishart ; posterior consistency ; convergence rate

Abstract

We consider Bayesian estimation of a precision matrix of order p, where p can be much larger than the available sample size n. We consider a banding structure in the model and induce a prior distribution on a banded precision matrix through a Gaussian graphical model, where an edge is present only when two vertices are within a given distance. We show that under a very mild growth condition and a proper choice of the order of graph, the posterior distribution based on the graphical model is consistent in the operator norm uniformly over a class of precision matrices. We conduct a simulation study to compare the Bayes estimator, the MLE based on the graphical model, and other standard estimators. We observe that the graphical model based estimators perform significantly better. We discuss a practical method of choosing the order of the graphical model using the marginal likelihood function.