The space of compatible full conditionals is a unimodular toric variety
The set of all m-tuples of compatible full conditional distributions on discrete random variables is an algebraic set whose defining ideal is a unimodular toric ideal. We identify the defining polynomials of these ideals with closed walks on a bipartite graph. Our algebraic characterization provides a natural generalization of the requirement that compatible conditionals have identical odds ratios and holds regardless of the patterns of zeros in the conditional arrays. (c) 2005 Elsevier Ltd. All rights reserved.
unimodular toric ideal odds ratios