Posterior Biomodality in the Balanced One-Way Random Effects Model

• Activity Number: 106
• Type: Contributed
• Date/Time: Monday, August 12, 2002 : 10:30 AM to 12:20 PM
• Sponsor: Section on Bayesian Stat. Sciences*
• Author(s): Jiannong Liu*+
• Affiliation(s): Minneapolis Medical Research Foundation
• Keywords: Bayesian Analysis ; Bimodality ; Hierarchical Model ; Conjugate Prior ; improper prior

Abstract

Recent computing advances have moved Bayesian analyses of complex hierarchical models out of seminar rooms, although our understanding of these models lags far behind. For example, bimodal posteriors are possible (Harville and Zimmermann 1996; O'Hagan 1976) and can cause embarrassing surprises (Wakefield's discussion on Hodges [1998]).

Although some authors have examined posterior multimodality for some specific models, multimodality is not well-characterized for any such model. This presentation characterizes the modality of the joint and some marginal posteriors for the balanced one-way random effects model (BOWREM) with conditionally conjugate prior for the variances and a flat prior on the mean. Posterior modality for the BOWREM may seem simple at first glance but, in fact, it is complex and sometimes bizarre. This presentation proves that the posteriors for the model can have at most two modes and describes when it is unimodal, when it is bimodal, and where is (are) the mode(s). The intuition for why and how a posterior become bimodal is given. Some examples show that bimodal posteriors appear often in practice even using almost noninformative priors.