Four Likelihoods of a Prior Density in the Empirical Bayes Model with Their Implications

• Activity Number: 263
• Type: Contributed
• Date/Time: Monday, August 10, 2015 : 2:00 PM to 3:50 PM
• Sponsor: Section on Bayesian Statistical Science
• Author(s): Takemi Yanagimoto*
• Companies: Institute of Statistical Mathematics
• Keywords: Bayes factor ; DIC ; information criterion ; non-informative prior

Abstract

Four likelihoods of a prior density in the empirical Bayes model are explored. They are defined in terms of four different predictive densities, which are classified by two mixing densities and also by the dual divergences. One of them is a specified version of the DIC. Other two relate with Bayes factor and with posterior Bayes factor, and the remaining one can be defined naively. They are compared under various models such as the exponential family and the finite mixture model. We observe that this approach elucidates close relations between the DIC and Bayes factor, and also that it combines coherently the likelihood in the empirical Bayes model and an information criterion in the frequentist approach. In fact, the likelihood ratio of two extreme models presents a clear interpretation on the AIC. Further, other likelihood inferential methods such as the likelihood ratio test are discussed. We find that a key issue in this study is how to define a non-informative prior in place of an unknown parameter in the frequentist model.