Multiway dependence in exponential family conditional distributions
Conditionally specified statistical models are frequently constructed from one-parameter exponential family conditional distributions. One way to formulate such a model is to specify the dependence structure among random variables through the use of a Markov random field (MRF). A common assumption on the Gibbsian form of the MRF model is that dependence is expressed only through pairs of random variables, which we refer to as the amp quot pairwise-only dependence amp;quot; assumption. Based on this assumption, J. Besag (1974, J. Roy. Statist. Soc. Ser. B 36, 192-225) formulated exponential family amp;quot;auto-models amp;quot; and showed the form that one-parameter exponential family conditional densities must take in such models. We extend these results by relaxing the pairwise-only dependence assumption, and we give a necessary form that one-parameter exponential family conditional densities must take under more general conditions of multiway dependence. Data on the spatial distribution of the European corn borer larvae are fitted using a model with Bernoulli conditional distributions and several dependence structures, including pairwise-only, three-way, and four-way dependencies. 2001 Academic Press.