A Two-step Metropolis Hastings Method for Bayesian Empirical Likelihood Computation with Application to Quantile Regression and Bayesian Model Selection

Empirical likelihood-based methods have been used under the Bayesian framework (BayesEL) in recent times. For statistical inference, these methods require efficient Markov chain Monte Carlo (MCMC) samplers for drawing observations from the parameter posterior distributions. However, the complex, especially non-convex, nature of the empirical likelihood support makes such MCMC algorithms harder to design. Such difficulties have restricted the use of BayesEL methods in many applications. In this article, we propose a two-step Metropolis-Hastings algorithm to sample from the BayesEL posteriors. Our proposal uses the current values of suitable subsets of the parameters and the estimating equations determining the underlying empirical likelihood to propose values of the remaining parameters. The proposed method is thus suitable for sampling from BayesEL posteriors in many complex problems, especially those with discontinuous estimating equations, e.g., simultaneous quantile regression. Furthermore, the proposed method easily extends to BayesEL model selection through a reversible jump Markov chain Monte Carlo procedure. Several illustrative, real-life applications of our proposed methods are presented.