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Algebraic geometry of Bayesian networks. (English) Zbl 1126.68102
Summary: We study the algebraic varieties defined by the conditional independence statements of Bayesian networks. A complete algebraic classification is given for Bayesian networks on at most five random variables. Hidden variables are related to the geometry of higher secant varieties.

MSC:
68W30Symbolic computation and algebraic computation
68T05Learning and adaptive systems
13P10Gröbner bases; other bases for ideals and modules
14N05Projective techniques (algebraic geometry)
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References:
[1] Barradas, I.; Solis, F.: Nonnegative rank. Int. math. J. 1, 601-610 (2002) · Zbl 1221.15023
[2] Catalano-Johnson, M.: The homogeneous ideals of higher secant varieties. J. pure appl. Algebra 158, 123-129 (2001) · Zbl 1006.14021
[3] Catalisano, M. V.; Geramita, A. V.; Gimigliano, A.: Ranks of tensors, secant varieties of Segre varieties and fat points. Linear algebra appl. 335, 263-285 (2002) · Zbl 1059.14061
[4] Cox, D.; Little, J.; O’shea, D.: Ideals, varieties and algorithms. Springer undergraduate texts in mathematics (1997)
[5] Decker, W.; Greuel, G. -M.; Pfister, G.: Primary decomposition: algorithms and comparisons. Algorithmic algebra and number theory, Heidelberg, 1997, 187-220 (1999) · Zbl 0932.13019
[6] Eisenbud, D.; Sturmfels, B.: Binomial ideals. Duke math. J. 84, 1-45 (1996) · Zbl 0873.13021
[7] Geiger, D.; Heckerman, D.; King, H.; Meek, C.: Stratified exponential families: graphical models and model selection. Ann. statist. 29, 505-529 (2001) · Zbl 1012.62012
[8] Geiger, D., Meek, C., Sturmfels, B., 2002. On the toric algebra of graphical models (manuscript) · Zbl 1104.60007
[9] Goodman, L.: Explanatory latent structure analysis using both identifiable and unidentifiable models. Biometrika 61, 215-231 (1974) · Zbl 0281.62057
[10] Grayson, D.; Stillman, M.: Macaulay2: A system for computation in algebraic geometry and commutative algebra. (1996)
[11] Greuel, G. -M.; Pfister, G.; Schönemann, H.: Singular 2.0: A computer algebra system for polynomial computations, university of Kaiserslautern. (2001)
[12] Harris, J., 1992. Algebraic Geometry: A First Course. Springer Graduate Texts in Mathematics · Zbl 0779.14001
[13] Lauritzen, S. L.: Graphical models. (1996) · Zbl 0907.62001
[14] Matúš, F.: Conditional independences among four random variables. III. final conclusion. Combin. probab. Comput. 8, 269-276 (1999) · Zbl 0941.60004
[15] Pearl, J.: Probabilistic reasoning in intelligent systems: networks of plausible inference. (1988) · Zbl 0746.68089
[16] Pistone, G.; Riccomagno, E.; Wynn, H.: Algebraic statistics: computational commutative algebra in statistics. (2001) · Zbl 0960.62003
[17] Settimi, R.; Smith, J. Q.: Geometry, moments and conditional independence trees with hidden variables. Ann. stat. 28, 1179-1205 (2000) · Zbl 1105.62321
[18] Settimi, R.; Smith, J. Q.: On the geometry of Bayesian graphical models with hidden variables. Proceedings of the fourteenth conference on uncertainty in artificial intelligence, 472-479 (1998)
[19] Shimoyama, T.; Yokoyama, K.: Localization and primary decomposition of polynomial ideals. J. symbolic comput. 22, 247-277 (1996) · Zbl 0874.13022
[20] Strassen, V.: Rank and optimal computation of generic tensors. Linear algebra appl. 52/53, 645-685 (1983) · Zbl 0514.15018
[21] Studený, M., 2001. On mathematical description of probabilistic conditional independence structures. Dr.Sc. Thesis, Prague, May. Posted at http://www.utia.cas.cz/user_data/studeny/studeny_home.html
[22] Sturmfels, B.: Gröbner bases and convex polytopes. American mathematical society, university lectures series 8 (1996) · Zbl 0856.13020
[23] Sturmfels, B.: Solving systems of polynomial equations. CBMS lectures series (2002) · Zbl 1101.13040