On the Beta Prime Prior for Scale Parameters in High-Dimensional Bayesian Regression Models

TL;DR

This paper introduces a Bayesian regression model using a beta prime prior for scale parameters, demonstrating near-minimax contraction rates in high-dimensional settings and providing adaptive hyperparameter estimation methods.

Contribution

It proposes a novel beta prime prior for scale parameters in high-dimensional Bayesian regression and develops data-driven hyperparameter estimation algorithms.

Findings

Achieves near-minimax posterior contraction rates for $p \,\gg\, n$

Provides adaptive hyperparameter estimation via MML

Demonstrates effectiveness through simulations and gene expression data analysis

Abstract

We study high-dimensional Bayesian linear regression with a general beta prime distribution for the scale parameter. Under the assumption of sparsity, we show that appropriate selection of the hyperparameters in the beta prime prior leads to the (near) minimax posterior contraction rate when $p \gg n$. For finite samples, we propose a data-adaptive method for estimating the hyperparameters based on marginal maximum likelihood (MML). This enables our prior to adapt to both sparse and dense settings, and under our proposed empirical Bayes procedure, the MML estimates are never at risk of collapsing to zero. We derive efficient Monte Carlo EM and variational EM algorithms for implementing our model, which are available in the R package NormalBetaPrime. Simulations and analysis of a gene expression data set illustrate our model's self-adaptivity to varying levels of sparsity and signal strengths.