Albert, James H.; Gupta, Arjun K. Bayesian estimation methods for 2\(\times 2\) contingency tables using mixtures of Dirichlet distributions. (English) Zbl 0546.62012 J. Am. Stat. Assoc. 78, 708-717 (1983). 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 Cited in 1 ReviewCited in 1 Document MSC: 62F15 Bayesian inference 62H17 Contingency tables 62H20 Measures of association (correlation, canonical correlation, etc.) Keywords:infinite population; categorical variables; multinomial distribution; prior distribution; misspecification of the tail portion of the prior; conjugate Dirichlet distribution; two-stage method; Approximate posterior means; credible regions PDFBibTeX XMLCite \textit{J. H. Albert} and \textit{A. K. Gupta}, J. Am. Stat. Assoc. 78, 708--717 (1983; Zbl 0546.62012) Full Text: DOI