# Revisiting High Dimensional Bayesian Model Selection for Gaussian Regression

## TL;DR

This paper analyzes Bayesian model selection in high-dimensional Gaussian regression, focusing on the interplay between prior choices, model complexity, and the asymptotic behavior of Bayes factors, with implications for consistency and false discovery control.

## Contribution

It introduces a novel analysis of the Zellner-Siow prior combined with a Poisson prior, revealing conditions for model selection consistency and false discovery control in high dimensions.

## Key findings

- Model selection consistency restricts true model dimension growth.
- Large covariate effects are needed to overcome complexity penalties.
- Asymptotic behavior of Bayes factors involves stable laws and heavy-tailed variables.

## Abstract

Model selection for regression problems with an increasing number of covariates continues to be an important problem both theoretically and in applications. Model selection consistency and mean structure reconstruction depend on the interplay between the Bayes factor learning rate and the penalization on model complexity. In this work, we present results for the Zellner-Siow prior for regression coefficients paired with a Poisson prior for model complexity. We show that model selection consistency restricts the dimension of the true model from increasing too quickly. Further, we show that the additional contribution to the mean structure from new covariates must be large enough to overcome the complexity penalty. The average Bayes factors for different sets of models involves random variables over the choices of columns from the design matrix. We show that a large class these random variables have no moments asymptotically and need to be analyzed using stable laws. We derive the domain of attraction for these random variables and obtain conditions on the design matrix that provide for the control of false discoveries.