On Computational Techniques for Bayesian Empirical Likelihood and Empirical Likelihood Based Bayesian Model Selection

Abstract

The posterior derived from the Bayesian empirical likelihood (BayEL) lacks analytical form and its support has complex geometry. Thus, efficient Markov chain Monte Carlo (MCMC) techniques are needed. In this thesis, two computational techniques are considered. We first consider Hamiltonian Monte Carlo (HMC) method, which takes advantage of the gradient of log Bayesian empirical likelihood posterior to guide the sampler in the non-convex posterior support. Due to the nature of the gradient, the HMC sampler would automatically draw samples within the support and rarely jumps out of it. The second method is a two-step Metropolis Hastings, which updates part of the parameters in the model based on the maximum empirical likelihood estimates given the remaining parameters. Our method is efficient in both fixed and varying dimensional state space. Another aspect considered in this thesis is the BayEL based Bayesian model selection. We propose an empirical likelihood based deviance information criterion (ELDIC), which has similar form to the classical deviance information criterion, but the definition of deviance now is based on empirical likelihood. The validity of ELDIC using as a criterion for Bayesian model selection is shown and the performance of ELDIC under different models is discussed.