Abstract
We consider Bayesian estimation of a $p\times p$ precision matrix, when $p$ can be much larger than the available sample size $n$. It is well known that consistent estimation in such ultra-high dimensional situations requires regularization such as banding, tapering or thresholding. 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. For a proper choice of the order of graph, we obtain the convergence rate of the posterior distribution and Bayes estimators based on the graphical model in the $L_{\infty}$-operator norm uniformly over a class of precision matrices, even if the true precision matrix may not have a banded structure. Along the way to the proof, we also compute the convergence rate of the maximum likelihood estimator (MLE) under the same set of condition, which is of independent interest. The graphical model based MLE and Bayes estimators are automatically positive definite, which is a desirable property not possessed by some other estimators in the literature. We also conduct a simulation study to compare finite sample performance of the Bayes estimators and the MLE based on the graphical model with that obtained by using a Cholesky decomposition of the precision matrix. Finally, we discuss a practical method of choosing the order of the graphical model using the marginal likelihood function.
Highlights
Estimating a covariance matrix or a precision matrix is one of the most important problems in multivariate analysis
Conventional estimators like the sample covariance matrix or maximum likelihood estimator behave poorly when the dimension is much higher than the sample size
Different regularization based methods have been proposed and developed in the recent years for dealing with high-dimensional data. These include banding, thresholding, tapering and penalization based methods to name a few; see, for example, [20, 16, 31, 2, 3, 17, 13, 28, 18, 27, 7, 5]. Most of these regularization based methods for high dimensional models impose a sparse structure in the covariance or the precision matrix, as in [2], where a rate of convergence has been derived for the estimator obtained by “banding” the sample covariance matrix, or by banding the Cholesky factor of the inverse sample covariance matrix, as long as n−1 log p → 0
Summary
Estimating a covariance matrix or a precision matrix (inverse covariance matrix) is one of the most important problems in multivariate analysis. These include banding, thresholding, tapering and penalization based methods to name a few; see, for example, [20, 16, 31, 2, 3, 17, 13, 28, 18, 27, 7, 5] Most of these regularization based methods for high dimensional models impose a sparse structure in the covariance or the precision matrix, as in [2], where a rate of convergence has been derived for the estimator obtained by “banding” the sample covariance matrix, or by banding the Cholesky factor of the inverse sample covariance matrix, as long as n−1 log p → 0. We consider Bayesian estimation of the precision matrix working with a G-Wishart prior induced by a Gaussian graphical model, which has a Markov property with respect to a decomposable graph G. Some auxiliary lemmas and their proofs are given in the Appendix
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