  Empirical likelihood is a popular nonparametric statistical tool that does
not require any distributional assumptions. In this paper, we explore the
possibility of conducting variable selection via Bayesian empirical likelihood.
We show theoretically that when the prior distribution satisfies certain mild
conditions, the corresponding Bayesian empirical likelihood estimators are
posteriorly consistent and variable selection consistent. As special cases, we
show the prior of Bayesian empirical likelihood LASSO and SCAD satisfies such
conditions and thus can identify the non-zero elements of the parameters with
probability tending to 1. In addition, it is easy to verify that those
conditions are met for other widely used priors such as ridge, elastic net and
adaptive LASSO. Empirical likelihood depends on a parameter that needs to be
obtained by numerically solving a non-linear equation. Thus, there exists no
conjugate prior for the posterior distribution, which causes the slow
convergence of the MCMC sampling algorithm in some cases. To solve this
problem, we propose a novel approach, which uses an approximation distribution
as the proposal. The computational results demonstrate quick convergence for
the examples used in the paper. We use both simulation and real data analyses
to illustrate the advantages of the proposed methods.

Methodology