Analysis of mixture models using expected posterior priors, with application to classificat.pdf

Analysis of Mixture Models Using Expected Posterior Priors, with Application to Classification of Gamma Ray Bursts JOSE M. PEREZ and JAMES O. BERGER Universidad Simon Bolivar, Venezuela and Duke University, USA Abstract: Consider observations distributed according to a mixture of compo- nent densities with different parameters. In the Bayesian framework, it is not possible to perform a 
  statistical analysis of the mixture using an improper prior for the component parameters, since the posterior distribution does not ex- ist. To overcome this difficulty, we propose use of the expected posterior prior approach of Perez and Berger (1999). Besides providing suitable default priors for general mixture models, a key advantage of the use of expected posterior priors is that they can be used in conjunction with Markov Chain Monte Carlo methods, even when the number of components is unknown. An application is considered involving Gamma Ray Bursts, modeled as arising from a bivariate normal mixture model with measurement errors on the observations. Keywords: MIXTURE MODELS; DEFAULT PRIORS; CLASSIFICATION; MCMC. 1. INTRODUCTION Mixture models have been used in situations including pseudo-parametric density estima- tion, clustering, change point problems and image analysis (McLachlan and Basford, 1985; Escobar and West, 1995; Roeder and Wasserman, 1997). In Bayesian analysis, mixture models are typically analyzed using Markov Chain Monte Carlo simulation (MCMC). See, for example, Diebolt and Robert (1994), Escobar and West (1995) and Richardson and Green (1998). Unfortunately, with mixture models it is typically not possible to perform default anal- ysis with standard noninformative priors. In this work, we demonstrate that the expected posterior prior approach (Perez and Berger, 1999) can be used to provide a default Bayesian analysis for this problem. In section 2, we define the expected posterior prior for mixture models. In section 3, a reversible jump MCMC sch