Consistent Bayesian Information Criterion Based on a Mixture Prior for Possibly High-Dimensional Multivariate Linear Regression Models

TL;DR

This paper introduces a new Bayesian information criterion for high-dimensional multivariate linear regression that combines features of AIC and BIC, ensuring consistency in variable selection across different sample sizes.

Contribution

It proposes a novel BIC based on a mixture prior that maintains asymptotic properties and consistency in high-dimensional settings, improving variable selection accuracy.

Findings

The new criterion is consistent in large-sample and high-dimensional asymptotics.

Numerical simulations show high probability of correctly selecting true variables.

Method outperforms traditional criteria in simulation studies.

Abstract

In the problem of selecting variables in a multivariate linear regression model, we derive new Bayesian information criteria based on a prior mixing a smooth distribution and a delta distribution. Each of them can be interpreted as a fusion of the Akaike information criterion (AIC) and the Bayesian information criterion (BIC). Inheriting their asymptotic properties, our information criteria are consistent in variable selection in both the large-sample and the high-dimensional asymptotic frameworks. In numerical simulations, variable selection methods based on our information criteria choose the true set of variables with high probability in most cases.