The problem of combining information related to I binomial experiments, each having a distinct probability of success theta(i), is considered. Instead of using a standard exchangeable prior for theta = (theta(1),..., theta(1)), we propose a more flexible distribution that takes into account various degrees of similarity among the theta(i)'s. Using ideas developed by Malec and Sedransk. we consider a partition g of the experiments and take the theta(i)'s belonging to the same partition subset to be exchangeable and the theta(i)'s belonging to distinct subsets to be independent. Next we perform Bayesian inference on theta conditional on g. Of course, one is typically uncertain about which partition to use, and so a prior distribution is assigned on a set of plausible partitions g. The final inference on theta is obtained by combining the conditional inferences according to the posterior distribution of g. The methodology adopted in this article offers a wide flexibility in structuring the dependence among the theta(i)'s. This allows the estimate of theta(i) to borrow strength from all other experiments according to an adaptive process governed by the data themselves. The method may be usefully applied to the analysis of binary response variables in the presence of categorical covariates. The latter are used to identify a collection of suitable partitions g, representing factor main effects and interactions, whose relevance will be summarized in the posterior distribution of g. Besides providing novel interpretations on the role played by the various factors, the procedure will also produce parameter estimates that may differ, sometimes in an appreciable manner, from those obtained using more traditional techniques. Finally, three real data sets are used to illustrate the methodology and compare it with other approaches, such as empirical Bayes (both parametric and nonparametric), logistic regression, and multiple shrinkage estimators.
