Bayesian Variable Selection for Linear Regression with the $\hat{I}º$-$G$ Priors

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 ${\hat{\mathbf{I}}}^2$, with each of the $g_j$'s, bounded between $0$ and $1$, on the diagonal serves as a stabilizer of the corresponding ${\hat{I}}_j^2$. 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.

19 pages, 3 figures, 1 table, 1 algorithm