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The space of compatible full conditionals is a unimodular toric variety
Aleksandra B Slavkovic, Seth Sullivant · 2004-05-04 · via math.ST updates on arXiv.org

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.

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