Default Bayesian analysis of mixture models

Chimei Shui, Purdue University

Abstract

Default Bayesian analysis has been very successful in dealing with most estimation and prediction problems. It is based on utilization of noninformative priors. For mixture models, however, default Bayesian analysis is typically not straightforward. The difficulty is that sophisticated noninformative priors are very difficult to determine, while simple choices will typically lead to improper posterior distributions. We advise two methods to overcome this difficulty. The first, called the iterative training sample classification (ITSC) method, evaluates posterior distributions and Bayes estimators of parameters based on a Monte Carlo method, utilizing improper priors. The main idea is to initially classify the sample into the population components of the mixture, and then use a "minimal training sample" from these populations to convert the improper prior to a proper prior. The resulting proper posterior is then used to reclassify the observations, and the analysis is repeated. Successive iterations are averaged geometrically, to yield the overall posterior distribution. The second approach we consider is called the iterative fractional Bayes factor approach, and uses a fixed point iterative scheme motivated by the ITSC method. It is substantially faster computationally, and yet converges to very similar answers. It also leads to the definition of an "intrinsic prior" for mixture models. Numerical examples, with both simulated data and real data, will be given. The determination of the number of components in a mixture model is also considered. Direct use of the intrinsic prior for this purpose is considered, but encounters difficulties. Thus we utilize the intrinsic prior through a modification of the fractional Bayes factor approach of O'Hagan (1995). This model selection criterion will be illustrated for univariate and bivariate normal mixtures. Artificial and real examples will be used for illustration.

Degree

Ph.D.

Advisors

Berger, Purdue University.

Subject Area

Statistics

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