Journal of Commutative Algebra

Positive margins and primary decomposition

Thomas Kahle, Johannes Rauh, and Seth Sullivant

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We study random walks on contingency tables with fixed marginals, corresponding to a (log-linear) hierarchical model. If the set of allowed moves is not a Markov basis, then tables exist with the same marginals that are not connected. We study linear conditions on the values of the marginals that ensure that all tables in a given fiber are connected. We show that many graphical models have the positive margins property, which says that all fibers with strictly positive marginals are connected by the quadratic moves that correspond to conditional independence statements. The property persists under natural operations such as gluing along cliques, but we also construct examples of graphical models not enjoying this property. We also provide a negative answer to a question of Engstr\"om, Kahle and Sullivant by demonstrating that the global Markov ideal of the complete bipartite graph $K_{3,3}$ is not radical.

Our analysis of the positive margins property depends on computing the primary decomposition of the associated conditional independence ideal. The main technical results of the paper are primary decompositions of the conditional independence ideals of graphical models of the $N$-cycle and the complete bipartite graph $K_{2,N-2}$, with various restrictions on the size of the nodes.

Article information

J. Commut. Algebra Volume 6, Number 2 (2014), 173-208.

First available in Project Euclid: 11 August 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 13P10: Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) 52B20: Lattice polytopes (including relations with commutative algebra and algebraic geometry) [See also 06A11, 13F20, 13Hxx]
Secondary: 05C81: Random walks on graphs 11P21: Lattice points in specified regions 60J22: Computational methods in Markov chains [See also 65C40] 62J12: Generalized linear models

Algebraic statistics Markov basis connectivity of fibers binomial primary decomposition


Kahle, Thomas; Rauh, Johannes; Sullivant, Seth. Positive margins and primary decomposition. J. Commut. Algebra 6 (2014), no. 2, 173--208. doi:10.1216/JCA-2014-6-2-173.

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