Posterior graph selection and estimation consistency for high-dimensional Bayesian DAG models

TLDR

Covariance estimation and selection in high‑dimensional data is a fundamental problem, and Gaussian DAG models impose sparsity in the Cholesky factor of the inverse covariance matrix, yet convergence and sparsity selection properties of Bayesian DAG‑Wishart priors have not been thoroughly investigated. The paper investigates a flexible class of DAG‑Wishart priors with multiple shape parameters. The authors analyze these priors within Bayesian inference for Gaussian DAG models. They prove strong graph selection consistency and posterior convergence rates when the number of variables grows subexponentially with sample size.

Abstract

Covariance estimation and selection for high-dimensional multivariate datasets is a fundamental problem in modern statistics. Gaussian directed acyclic graph (DAG) models are a popular class of models used for this purpose. Gaussian DAG models introduce sparsity in the Cholesky factor of the inverse covariance matrix, and the sparsity pattern in turn corresponds to specific conditional independence assumptions on the underlying variables. A variety of priors have been developed in recent years for Bayesian inference in DAG models, yet crucial convergence and sparsity selection properties for these models have not been thoroughly investigated. Most of these priors are adaptations/generalizations of the Wishart distribution in the DAG context. In this paper, we consider a flexible and general class of these “DAG-Wishart” priors with multiple shape parameters. Under mild regularity assumptions, we establish strong graph selection consistency and establish posterior convergence rates for estimation when the number of variables $p$ is allowed to grow at an appropriate subexponential rate with the sample size $n$.