Bayesian Inference in Incomplete Multi-Way Tables

Summary We describe and illustrate approaches to Bayesian inference in multi-way contingency ta-bles for which partial information, in the form of subsets of marginal totals, is available.In such problems, interest lies in questions of inference about the parameters of modelsunderlying the table together with imputation for the individual cell entries. We discussquestions of structure related to the implications for inference on cell counts arising fromassumptions about log-linear model forms, and a class of simple and useful prior distribu-tions on the parameters of log-linear models. We then discuss \local move" and \globalmove" Metropolis-Hastings simulation methods for exploring the posterior distributions forparameters and cell counts, focusing particularly on higher-dimensional problems. As a by-product, we note potential uses of the \global move" approach for inference about numbersof tables consistent with a prescribed subset of marginal counts. Illustration and compar-ison of MCMC approaches is given, and we conclude with discussion of areas for furtherdevelopments and current open issues.Some key words: Bayesian inference; Disclosure limitation; Fixed margins problem; Imputation; Log-linear models; Markov basis; Markov chain Monte Carlo; Missing data.2