Abstract
The binomial ideal associated with the intersection axiom of conditional probability is shown to be radical and is expressed as an intersection of toric prime ideals. This solves a problem in algebraic statistics posed by Cartwright and Engström.
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Drton, M., Sturmfels, B., Sullivant, S.: Lectures on Algebraic Statistics. Oberwolfach Seminars, vol. 39. Springer, Berlin (2009)
Eisenbud, D., Sturmfels, B.: Binomial ideals. Duke Math. J. 84, 1–45 (1996)
Grayson, D.R., Stillman, M.: Macaulay2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2/
Greuel, G.-M., Pfister, G., Schönemann, H.: Singular 3.0—a computer algebra system for polynomial computations. In: Kerber, M., Kohlhase, M. (eds.) Symbolic Computation and Automated Reasoning, The Calculemus-2000 Symposium, pp. 227–233 (2001)
Greuel, G.-M., Pfister, G.: primdec.lib, a Singular 3.0 library for computing the primary decomposition and radical of ideals (2005)
Herzog, J., Hibi, T., Hreinsdóttir, F., Kahle, T., Rauh, J.: Binomial edge ideals and conditional independence statements. arXiv:0909.4717
Kahle, T.: Binomials.m2, code for binomial primary decomposition in Macaulay2. http://personal-homepages.mis.mpg.de/kahle/bpd/index.html
Sturmfels, B.: Gröbner bases of toric varieties. Tōhoku Math. J. 43, 249–261 (1991)
Stanley, R.P.: Enumerative Combinatorics, vol. 2. Cambridge Studies in Advanced Mathematics, vol. 62. Cambridge University Press, Cambridge (1997)
Studený, M.: Probabilistic Conditional Independence Structures, Information Science and Statistics. Springer, New York (2005)
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Fink, A. The binomial ideal of the intersection axiom for conditional probabilities. J Algebr Comb 33, 455–463 (2011). https://doi.org/10.1007/s10801-010-0253-5
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DOI: https://doi.org/10.1007/s10801-010-0253-5