Bayesian Variable Selection for Linear Regression with the $\kappa$-$G$ Priors

TL;DR

This paper proposes a novel Bayesian variable selection method for linear regression using $oldsymbol{ ext{kappa}}$-$oldsymbol{ ext{G}}$ priors, which stabilizes coefficients with a diagonal matrix and employs Metropolis-within-Gibbs sampling.

Contribution

It introduces a new prior-based variable selection approach that is independent of indicator methods, with proven properties and a practical sampling algorithm.

Findings

Method effectively distinguishes promising and unpromising variables.

Simulation results demonstrate successful variable selection.

Provides asymptotic properties under orthogonality.

Abstract

In this paper, we introduce a new methodology for Bayesian variable selection in linear regression that is independent of the traditional indicator method. A diagonal matrix $\mathbf{G}$ is introduced to the prior of the coefficient vector $\boldsymbol{\beta}$, with each of the $g_j$'s, bounded between $0$ and $1$, on the diagonal serves as a stabilizer of the corresponding $\beta_j$. Mathematically, a promising variable has a $g_j$ value that is close to $0$, whereas the value of $g_j$ corresponding to an unpromising variable is close to $1$. This property is proven in this paper under orthogonality together with other asymptotic properties. Computationally, the sample path of each $g_j$ is obtained through Metropolis-within-Gibbs sampling method. Also, in this paper we give two simulations to verify the capability of this methodology in variable selection.