Compatibility of discrete conditional distributions with structural zeros

A general algorithm is provided for determining the compatibility among full conditionals of discrete random variables with structural zeros. The algorithm is scalable and it can be implemented in a fairly straightforward manner. A MATLAB program is included in the Appendix and therefore, it is now feasible to check the compatibility of multi-dimensional conditional distributions with constrained supports. Rather than the linear equations in the restricted domain of Arnold et al. (2002) [11] Tian et al. (2009) [16], the approach is odds-oriented and it is a discrete adaptation of the compatibility check of Besag (1994) [17]. The method naturally leads to the calculation of a compatible joint distribution or, in the absence of compatibility, a nearly compatible joint distribution. Besag's [5] factorization of a joint density in terms of conditional densities is used to justify the algorithm.