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

In recent times empirical likelihood has been widely applied under Bayesian framework. Markov chain Monte Carlo (MCMC) methods are frequently employed to sample from the posterior distribution of the parameters of interest. However, complex, especially non-convex nature of the likelihood support erects enormous hindrances in choosing an appropriate MCMC algorithm. Such difficulties have restricted the use of Bayesian empirical likelihood (BayesEL) based methods in many applications. In this article, we propose a two-step Metropolis Hastings algorithm to sample from the BayesEL posteriors. Our proposal is specified hierarchically, where the estimating equations determining the empirical likelihood are used to propose values of a set of parameters depending on the proposed values of the remaining parameters. Furthermore, we discuss Bayesian model selection using empirical likelihood and extend our two-step Metropolis Hastings algorithm to a reversible jump Markov chain Monte Carlo procedure to sample from the resulting posterior. Finally, several applications of our proposed methods are presented.