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Conditionally specified statistical models are frequently constructed from conditional one-parameter exponential family 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. When this is done, a common assumption is that dependence is expressed only through pairs of random variables, the 'pairwise-only dependence' assumption. Using a Markov random field structure and the pairwise-only dependence assumption, Besag (1974) formulated exponential family 'auto-models', and showed the form that conditional one-parameter exponential family densities must have in such models. We extend those results under relaxation of the pairwise-only dependence assumption, and give a necessary form for conditional one-parameter exponential family densities under more general conditions of multiway dependence.

Specification of dependence structures and simulation-based estimation for conditionally specified statistical models