Bayesian Methods for Two-way Contingency Tables

In this paper, we consider Bayesian estimation and model determination for two-way contingency tables. The analysis is based on partitioning the parameter space of a saturated log-linear model into parameter subspaces identiied by considering the natural symmetries of the data. Prior distributions may then be chosen to respect invariance restrictions required by the available prior information. We focus particular attention on the case of a square contingency table, where prior information is invariant to permutation of the row (column) category levels, and to interchanging the row and column variable. The resulting class of invariant log-linear models includes many familiar models for this kind of data (Symmetry, Quasi-Symmetry, Quasi-Independence etc.) Posterior inference is reasonably straightforward, and inference related to model determination may be presented either through posterior probabilities of individual models, or by considering marginal posterior distributions within the invariant parameter spaces. The analysis of an intermarriage table is presented as an example.