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Conditionally specified continuous distributions
Department of Mathematical Sciences, Rutgers University, Camden, New Jersey 08102, U.S.A. yuwang{at}crab.rutgers.edu
Department of Biostatistical Sciences, Wake Forest University School of Medicine, Winston-Salem, North Carolina 27157, U.S.A. eip{at}wfubmc.edu
Received for publication 1 November 2006. Revision received 1 February 2008.
A distribution is conditionally specified when its model constraints are expressed conditionally. For example, Besag's (1974) spatial model was specified conditioned on the neighbouring states, and pseudolikelihood is intended to approximate the likelihood using conditional likelihoods. There are three issues of interest: existence, uniqueness and computation of a joint distribution. In the literature, most results and proofs are for discrete probabilities; here we exclusively study distributions with continuous state space. We examine all three issues using the dependence functions derived from decomposition of the conditional densities. We show that certain dependence functions of the joint density are shared with its conditional densities. Therefore, two conditional densities involving the same set of variables are compatible if their overlapping dependence functions are identical. We prove that the joint density is unique when the set of dependence functions is both compatible and complete. In addition, a joint density, apart from a constant, can be computed from the dependence functions in closed form. Since all of the results are expressed in terms of dependence functions, we consider our approach to be dependence-based, whereas methods in the literature are generally density-based. Applications of the dependence-based formulation are discussed.
Key Words: Compatibility • Gibbs sampler • Global dependence function • Interaction • Local dependence function • Log-linear decomposition • Mixed parameterization • Moebius inversion formula • Pseudolikelihood • Subset lattice
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