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

In this article we develop a new methodology for Bayesian variable selection in multiple linear regression that is independent of the standard indicator vector method. Serving as an extension of Zellner's $g$-prior, we extend the original scalar $g$ to a diagonal matrix $G$ that controls the stability of the prior on the coefficients $\boldsymbol{\beta}$, and each of the elements $g_j$ controls the stability of its corresponding dimension. From the Metropolis-within-Gibbs sampling method, the posterior values of $G$ are sampled and those promising variables tend to have a $g_j$'s that are close to $0$. Thus the promising variables are chosen based on the posterior of $g_j$. As each of the $g_j$'s is a stabilizer of its own dimension, the $1-g_j$ values imitates the posterior inclusion probability (PIP) as in the standard methodology.