# Prior Induction in Log-linear Models for General Contingency Table Analysis

### Ruth King and Stephen P. Brooks

### University of Cambridge

## Summary

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 the Bayesian analysis of these models, and in particular model
selection and averaging, has become feasible.
Coupled with this is the desire to construct and use suitably flexible
prior structures which allow efficient computation whilst 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
log-normal 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 2x2x2 contingency table.

### Keywords: Bayesian analysis, contingency table, multivariate normal, prior
elicitation.

Appeared as King, R. and Brooks, S.P. (2001) "Prior Induction in Log-Linear
Modelling".* Annals of Statistics * **29** pp 715-747