Bayesian interference for Log-Linear models

Abstract

The conditional Dirichlet distribution, which has the attractive property that its parameters may be interpreted as prior cell counts, is introduced. This prior is useful for both reference analyses, where small prior values are used, and as an informative prior, where (hypothetical) prior cell counts may be available. The conditional Dirichlet is shown to be equivalent to a hyper Dirichlet density (which admits straightforward analyses) for decomposable log-linear models. Hence a natural extension of the hyper Dirichlet distribution to non-decomposable models is obtained.

The conditional Dirichlet distribution is not tractable in general, so Monte Carlo and other approximation methods are required. Gibbs sampling is applied to obtain samples from prior and posterior conditional Dirichlet distributions. The sampler is found to mix well, producing samples which are not highly dependent.

Laplace's method is used for the approximation of integrals, although it is found to perform poorly where prior parameters take small values. However, accurate results may be obtained for the posterior analysis of datasets where cell counts are large. The method of bridge sampling is applied to the problem of determining the normalising constants for conditional Dirichlet distributions. The sampler is found to produce good results, even when prior parameters take small values, and this is illustrated by application to several examples.

Jeffrey's prior, which is a reference prior by definition, is considered, and an explicit expression is presented for the Jeffreys' prior for a decomposable log-linear model. In many cases, this is found to be a product of independent Dirichlet distributions for the parameters of a particular decomposition of the model. For other decomposable models, where the normalising constant for Jeffreys' prior is not directly available, the method of bridge sampling is again applied, and found to produce accurate results. The Monte Carlo samples needed are obtained using Metropolis Hastings sampling.

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