# A novel Bayesian approach for variable selection in linear regression models

## TL;DR

This paper introduces a new Bayesian method for variable selection in linear regression that incorporates prior beliefs, uses a stable g-prior, and employs an advanced MCMC algorithm, demonstrating competitive performance and consistency.

## Contribution

It presents a hierarchical Bayesian framework with a novel MCMC algorithm for variable selection, allowing direct prior specification and ensuring numerical stability.

## Key findings

- Performs at least as well as existing methods on real and simulated data.
- Proves consistency of the variable selection approach under certain conditions.
- Introduces a stable g-prior with ridge parameter for improved numerical stability.

## Abstract

We propose a novel Bayesian approach to the problem of variable selection in multiple linear regression models. In particular, we present a hierarchical setting which allows for direct specification of a-priori beliefs about the number of nonzero regression coefficients as well as a specification of beliefs that given coefficients are nonzero. To guarantee numerical stability, we adopt a $g$-prior with an additional ridge parameter for the unknown regression coefficients. In order to simulate from the joint posterior distribution an intelligent random walk Metropolis-Hastings algorithm which is able to switch between different models is proposed. Testing our algorithm on real and simulated data illustrates that it performs at least on par and often even better than other well-established methods. Finally, we prove that under some nominal assumptions, the presented approach is consistent in terms of model selection.