Bayesian inference for square contingency tables

Inference for multivariate categorical data often proceeds by selecting a log-linear model from a set of competing models or, in a Bayesian approach, by averaging inferences over the set, weighted by posterior probabilities. In this paper, we use permutation invariance as a criterion for constructing a set of models for this purpose, for the common situation when the data form a ‘square’ contingency table, representing a cross-classification by two categorical variables with identical categories. We consider log-linear models which are invariant under certain groups of permutations of the cells of a multiway contingency table. We present permutation invariant log-linear models for a number of different contingency table structures and show how to construct invariant prior distributions for the model parameters.