Markov Chain Monte Carlo Implementation of Empirical Bayes and Likelihood Inference

• Activity Number: 414
• Type: Contributed
• Date/Time: Tuesday, August 5, 2014 : 2:00 PM to 3:50 PM
• Sponsor: Section on Bayesian Statistical Science
• Author(s): Yeonhee Park*+ and Hani Doss
• Companies: University of Florida and University of Florida
• Keywords: Bayesian statistics ; hyperparameter ; Empirical Bayes ; the maximizer of marginal likelihood ; Markov Chain Monte Carlo ; empirical process

Abstract

In Bayesian statistics, the prior is typically chosen from a family of distributions indexed by some hyperparameter, and the choice of this hyperparameter is important, as it affects subsequent inference. To select it, ideally we would form the marginal likelihood of the data as a function of the hyperparameter; the Empirical Bayes choice is, by definition, the value of the hyperparameter that maximizes this marginal likelihood. Unfortunately, in all but the simplest examples, the marginal likelihood is not available in closed form. However, it turns out that typically, the entire likelihood surface can be estimated by Markov chain Monte Carlo, using a single Markov chain run. We present a method for forming point estimates and confidence sets for the maximizer of the marginal likelihood. The theoretical basis for the method is established by using tools from the theory of empirical processes.