Necessary and Sufficient Conditions for High-Dimensional Posterior Consistency under $g$-Priors

TL;DR

This paper establishes necessary and sufficient conditions for high-dimensional posterior consistency under g-priors, extending to hierarchical and empirical Bayesian models with growing regressors and using the sup norm.

Contribution

It provides new theoretical conditions for posterior consistency in high-dimensional Bayesian regression models with g-priors, including extensions to hierarchical and empirical Bayesian frameworks.

Findings

Conditions for posterior consistency under g-priors are characterized.

Consistency results are extended to models with a growing number of regressors.

Posterior consistency is analyzed under the sup vector norm.

Abstract

We examine necessary and sufficient conditions for posterior consistency under $g$-priors, including extensions to hierarchical and empirical Bayesian models. The key features of this article are that we allow the number of regressors to grow at the same rate as the sample size and define posterior consistency under the sup vector norm instead of the more conventional Euclidean norm. We consider in particular the empirical Bayesian model of George and Foster (2000), the hyper-$g$-prior of Liang et al. (2008), and the prior considered by Zellner and Siow (1980).