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Bayesian estimation methods for 2\(\times 2\) contingency tables using mixtures of Dirichlet distributions. (English) Zbl 0546.62012

When data from an infinite population are collected and classified with respect to two categorical variables A and B each having two levels, the cell counts \(X_{ij}\), \(i=1,2\) and \(j=1,2\) possess a multinomial distribution with cell probabilities \({\underset \tilde{}\theta }=\{\theta_{11},\theta_{12},\theta_{21},\theta_{22}\}\). This paper is concerned with the Bayesian estimation of \({\underset \tilde{}\theta }.\)
Consider the measure of association \(\alpha =(\theta_{11}\theta_{22})/(\theta_{12}\theta_{21})\). The prior beliefs of \(\alpha\) are usually vague, but one may be able to specify the location of the middle portion of its prior distribution. The authors aim at using a prior distribution that reflects information about the central portion of the prior on \(\alpha\) and yields posterior estimates of \({\underset \tilde{}\theta }\) that are robust with respect to misspecification of the tail portion of the prior. The method of imputing a prior by assuming the conjugate Dirichlet distribution for \({\underset \tilde{}\theta }\) is not satisfactory, since it does not incorporate the user’s ignorance of the tail portion of the prior distribution of \(\alpha\).
The authors suggest a two-stage method for Bayesian inference about the parameters \({\underset \tilde{}\theta }\). At stage one, the distribution of \({\underset \tilde{}\theta }\) is assumed to be of the conjugate Dirichlet form with some parameters of this distribution being unknown. At the second stage, the unknown parameters are given a completely specified distribution. Approximate posterior means and credible regions are developed for the case when a priori, the two variables are independent.
Reviewer: U.D.Naik

MSC:

62F15 Bayesian inference
62H17 Contingency tables
62H20 Measures of association (correlation, canonical correlation, etc.)
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