Bayesiam simultaneous estimation for several multinomial distributions

Abstract

A Bayesian method is proposed for estimating the cell probabilities of several multinomial distributions. Parameters of different distributions are taken to be a priori exchangeable. The prior specification is based upon mixtures of a hierarchical distribution, referred to as the multivariate “Dirichlet-Dirichlet” distribution. The analysis is facilitated by a multinomial approximation relating to the multinomial-Dirichlet distribution. The posterior estimates depend upon measures of entropy for the various distributions and shrink the individual observed proportions towards values obtained by pooling the data across the distributions. As well as incorporating prior information they are particularly useful when some of the cell frequencies are zero. We use them to investigate a numerical classification of males of various vocations, according to cause of death.

Keywords:

• Dirichlet-Dirichlet
• mixtures
• contingency tables
• exchangeability
• James-Stein
• posterior means
• zero cell frequencies
• measures of entropy
• multinomial-Dirichlet
• shrinkage of estimates