Prior Induction in Log-Linear Models for General Contingency Table Analysis

Log-linear modelling plays an important role in many statistical applications, particularly in the analysis of contingency table data. With the advent of powerful new computational techniques such as reversible jump MCMC, Bayesian analyses of these models, and in particular model selection and averaging, have become feasible. Coupled with this is the desire to construct and use suitably flexible prior structures which allow efficient computation while facilitating prior elicitation. The latter is greatly improved in the case where priors can be specified on interpretable parameters about which relevant experts can express their beliefs.

In this paper, we show how the specification of a general multivariate normal prior on the log-linear parameters induces a multivariate lognormal prior on the corresponding cell counts of a contingency table. We derive the parameters of this distribution in an explicit practical form and state the corresponding mean and covariances of the cell counts. We discuss the importance of these results in terms of applying both uninformative and informative priors to the model parameters and provide an illustration in the context of the analysis of a 2(3) contingency table.