The space of compatible full conditionals is a unimodular toric variety

  • Authors:

  • Aleksandra B. Slavkovic;Seth Sullivant

  • Affiliations:

  • Department of Statistics, Pennsylvania State University, United States;Department of Mathematics, University of California, Berkeley, United States

  • Venue:
  • Journal of Symbolic Computation
  • Year:
  • 2006

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Abstract

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.