Journal of Commutative Algebra

Positive margins and primary decomposition

Thomas Kahle, Johannes Rauh, and Seth Sullivant

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Abstract

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

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

Dates
First available in Project Euclid: 11 August 2014

Permanent link to this document
https://projecteuclid.org/euclid.jca/1407790529

Digital Object Identifier
doi:10.1216/JCA-2014-6-2-173

Mathematical Reviews number (MathSciNet)
MR3249835

Zentralblatt MATH identifier
1375.13047

Subjects
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

Keywords
Algebraic statistics Markov basis connectivity of fibers binomial primary decomposition

Citation

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. https://projecteuclid.org/euclid.jca/1407790529


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References

  • 4ti2-A software package for algebraic, geometric and combinatorial problems on linear spaces, available at www.4ti2.de, 2007.
  • J. Besag, Spatial interaction and the statistical analysis of lattice systems, J. Roy. Stat. Soc. 36 (1974), 192-236.
  • René Birkner, Polyhedra: A package for computations with convex polyhedral objects, J. Software Alg. Geom. 1 (2009), 11-15.
  • Nicolas Bourbaki, Éléments de mathématique, in Algèbre, Hermann, 1950.
  • Florentina Bunea and Julian Besag, MCMC in $i \times j \times k$ contingency tables, in Monte Carlo methods, Neal Madras, ed., Fields Inst. Comm. 26, AMS and Fields Institute, 2000.
  • Yuguo Chen, Ian H. Dinwoodie and Seth Sullivant, Sequential importance sampling for multiway tables, Ann. Stat. 34 (2006), 523-545.
  • Yuguo Chen, Ian Dinwoodie and Ruriko Yoshida, Markov chains, quotient ideals, and connectivity with positive margins, in Algebraic and geometric methods in statistics, Pablo Gibilisco, Eva Riccomagno, Maria Piera Rogantin and Henry P. Wynn, eds., Cambridge University Press, Cambridge, 2010.
  • József Dénes and A.D. Keedwell, Latin squares and their applications, Academic Press, New York, 1974.
  • Mike Develin and Seth Sullivant, Markov bases of binary graph models, Ann. Comb. 7 (2003), 441-466.
  • Persi Diaconis, David Eisenbud and Bernd Sturmfels, Lattice walks and primary decomposition, in Mathematical essays in honor of Gian-Carlo Rota, B. Sagan and R. Stanley, eds., Progr. Math. 161, Birkhauser, Boston, 1998.
  • Persi Diaconis and Bernd Sturmfels, Algebraic algorithms for sampling from conditional distributions, Ann. Stat. 26 (1998), 363–397.
  • Mathias Drton, Bernd Sturmfels and Seth Sullivant, Lectures on algebraic statistics, Oberwolfach Sem. 39, Birkhäuser, Springer, Berlin, 2009.
  • David Eisenbud and Bernd Sturmfels, Binomial ideals, Duke Math. J. 84 (1996), 1-45.
  • Alexander Engström, Thomas Kahle and Seth Sullivant, Multigraded commutative algebra of graph decompositions, J. Alg. Comb. 39 (2014), 335-372.
  • Dan Geiger, Christopher Meek and Bernd Sturmfels, On the toric algebra of graphical models, Ann. Stat. 34 (2006), 1463-1492.
  • Daniel R. Grayson and Michael E. Stillman, Macaulay$2$, A software system for research in algebraic geometry, Available at http://www.math.uiuc.edu/Macaulay2/.
  • Jürgen Herzog, Takayuki Hibi, Freyja Hreinsdóttir, Thomas Kahle and Johannes Rauh, Binomial edge ideals and conditional independence statements, Adv. Appl. Math. 45 (2010), 317-333.
  • Thomas Kahle, Decompositions of binomial ideals, J. Software Alg. Geom. 4 (2012), 1-5.
  • –––, GraphBinomials, A library for walks on graphs on monomials, available from https://github.com/tom111/GraphBinomials, 2012.
  • Thomas Kahle and Ezra Miller, Decompositions of commutative monoid congruences and binomial ideals, 2011, arXiv:1107.4699.
  • Steffen L. Lauritzen, Graphical models, Oxford Stat. Sci. Ser., Oxford University Press, Oxford, 1996.
  • Jesús A. De Loera and Shmuel Onn, Markov bases of three-way tables are arbitrarily complicated, J. Symb. Comp. 41 (2006), 173-181.
  • Peter N. Malkin, Truncated Markov bases and Gröbner bases for integer programming, manuscript, 2006.
  • Johannes Rauh, Generalized binomial edge ideals, Adv. Appl. Math. 50 (2013), 409-414.
  • Johannes Rauh and Nihat Ay, Robustness canalyzing functions and systems design, Theor. Biosci. 133 (2014), 63-78.
  • Johannes Rauh and Thomas Kahle, The Markov bases database, http:/\!\!/markov-bases.de.
  • Ronald C. Read and Robin J. Wilson, An atlas of graphs, Clarendon Press, New York, 1998.
  • Bernd Sturmfels, Gröbner bases of toric varieties, Tōhoku Math. J. 43 (1991), 249-261.
  • –––, Gröbner bases and convex polytopes, University Lecture Series 8, American Mathematical Society, Providence, RI, 1996.
  • –––, Solving systems of polynomial equations, CBMS 97, American Mathematical Society, Providence, 2002.
  • Seth Sullivant, Toric fiber products, J. Algebra 316 (2007), 560-577.
  • Irena Swanson and Amelia Taylor, Minimal primes of ideals arising from conditional independence statements, J. Alg. 392 (2013), 299-314. \noindentstyle