The space of compatible full conditionals is a unimodular toric variety. (English) Zbl 1125.60003
Summary: 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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