This page collects some references to works about Markov bases. The selection is a bit arbitrary, and if you feel
that we have missed an important fact or reference, please tell us.
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4ti2 team. 4ti2 -- A software package for algebraic, geometric and
combinatorial problems on linear spaces. Available at www.4ti2.de.
- Most computations of Markov bases are done using 4ti2.
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M. Develin, S. Sullivant. Markov Bases of Binary Graph Models. Annals of Combinatorics 7 (2003), 441-466.
- It is stated, that a graph model has Markov degree 2 iff its graph is a forest.
- This paper shows that the Markov degree of the graph models of CN and K(2,N) is 4.
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P. Diaconis, B. Sturmfels. Algebraic algorithms for sampling from conditional distributions. Annals of
Statistics, 26 (1998), 363-397.
- The idea to use Markov bases in statistics was first published here.
- D. Geiger, C. Meek, B. Sturmfels. On the toric algebra of graphical models. Annals of Statistics 34, Nr 3 (2006), 1463-1492.
- This paper shows that the Markov degree of a graphical model is 2 iff the graph is decomposable.
- S. Hoşten, S. Sullivant. Gröbner bases and polyhedral geometry of reducible and cyclic models. Journal of Combinatorial Theory Ser. A 100 (2002), 277-301.
- This paper shows that the Markov degree of a graphical model is 2 iff the graph is decomposable.
- V. Csiszàr. Conditional independencs relations and log-linear models for random matchings. Acta Mathematica Hungarica, online first.
- L-decomposable and bidecomposable random permutations are discussed.
- J.I. García-García, M.A. Moreno-Frías, A. Vigneron-Tenorio. On the decomposable semigroups and their
applications in Algebraic Statistics.
Arxiv 1006.2557v2 (2011).
- This article decides uniqueness of Markov bases for decomposable models.
- Daniel Irving Bernstein and Seth Sullivant. Unimodular binary hierarchical models. Journal of Combinatorial Theory, Series B 123 (2017): 97-125.
- This paper shows unimodularity and thus normality and uniqueness of some Markov bases in the database.
- Daniel Irving Bernstein and Christopher O'Neill. Unimodular hierarchical models and their Graver bases. Journal of Algebraic Statistics 8 (2017): 29-43.
- This paper shows unimodularity and thus normality and uniqueness of some Markov bases in the database.
