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

Markov chain Monte Carlo (MCMC) methods are frequently employed to sample from the posterior distribution of the parameters of interest. 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. In recent years, empirical likelihood (Owen, 1988; Qin & Lawless, 1994) based procedures have been frequently used under Bayesian framework. Such procedures specify a statistical model through unbiased estimating equations, without requiring a declaration of the data distribution. The likelihood is estimated from the empirical distribution function computed under constraints imposed by these estimating equations. The estimated likelihood is then used to define a posterior. The validity of empirical and similar likelihoods for Bayesian inference has been a topic of extensive discussion (Monahan & Boos, 1992; Lazar, 2003; Fang & Mukerjee, 2006; Corcoran, 1998). Alternative likelihoods like Bayesian exponential tilted empirical likelihood (BETEL) (Schennach, 2005) have been proposed and justified using basic probabilistic arguments.