Bayes and Empirical Bayes Procedures for Comparing Parameters

Abstract

The number of successes out of the total number of trials is observed for each of a set of N independent binomial experiment's and the associated success probabilities are compared. Ranking procedures for comparing these success probabilities are constructed under circumstances where the number of trials for each of the experiments may differ by a large amount. A loss function is constructed that represents an appropriate measure of distance between a given ranking of the success probabilities and their true ranking. Bayes and empirical Bayes procedures based on this loss function are proposed and compared with “adapted ranking procedures” that arise through Bayes, empirical Bayes, or maximum likelihood estimation of the success probabilities. Bayes and empirical Bayes ranking procedures are shown to be transitive. Empirical Bayes ranking procedures are shown to arise from maximizing a function of the prior parameters that measures the amount of confidence enjoyed by various Bayes procedures. These results are applied to comparing the prevalence of toxoplasmosis for different small communities in El Salvador and comparing the batting averages of baseball players in the World Series.

Key Words:

• Hierarchical Bayes
• Posterior risk
• Tournament procedures
• Transitivity