Algebraic geometry of Gaussian Bayesian networks

Advances in Applied Mathematics

Volumn 40, Issue 4, 2008, Pages 482-513

  Sullivant, Seth ^a  

^a HARVARD UNIVERSITY   (United States)

Author keywords

Algebraic statistics; Bayesian network; Graphical model; Gr bner basis; Multivariate Gaussian

Indexed keywords

COMPUTATIONAL GEOMETRY; COVARIANCE MATRIX; GRAPHIC METHODS; MULTIVARIABLE CONTROL SYSTEMS; RANDOM VARIABLES; STATISTICS;

ALGEBRAIC STATISTICS; GRAPHICAL MODELS; GROBNER BASIS; MULTIVARIATE GAUSSIAN; UPSTREAM RANDOM VARIABLES;

BAYESIAN NETWORKS;

EID: 42049122767     PISSN: 01968858     EISSN: 10902074     Source Type: Journal    
DOI: 10.1016/j.aam.2007.04.004     Document Type: Article

References (20)

1. Bickel P., and Doksum K. Mathematical Statistics, vol. 1 (2001), Prentice Hall, London
2. Cox D., Little J., and O'Shea D. Ideals, Varieties, and Algorithms (1997), Springer, New York
3. Drton M., Sturmfels B., and Sullivant S. Algebraic factor analysis: Tetrads, pentads, and beyond. Probab. Theory Related Fields 138 3-4 (2007) 463-493
4. Durbin R., Eddy S., Krogh A., and Mitchison G. Biological Sequence Analysis (1999), Cambridge Univ. Press, Cambridge
5. Felsenstein J. Inferring Phylogenies (2004), Sinauer Associates, Sunderland, MA
6. L. Garcia, Polynomial constraints of Bayesian networks with hidden variables, preprint, 2006
7. Garcia L., Stillman M., and Sturmfels B. Algebraic geometry of Bayesian networks. J. Symbolic Comput. 39 (2005) 331-355
8. Gessel I., and Viennot G. Binomial determinants, paths, and hook length formulae. Adv. Math. 58 3 (1985) 300-321
9. Grayson D., and Stillman M. Macaulay 2, a software system for research in algebraic geometry. available at. http://www.math.uiuc.edu/Macaulay2/
10. Hochster M. Rings of invariants of tori, Cohen-Macaulay rings generated by monomials, and polytopes. Ann. of Math. 96 (1972) 318-337
11. Knutson A., and Miller E. Gröbner geometry of Schubert polynomials. Ann. of Math. (2) 161 3 (2005) 1245-1318
12. Landsberg J.M., and Manivel L. On the ideals of secant varieties of Segre varieties. Found. Comput. Math. 4 4 (2004) 397-422
13. Lauritzen S. Graphical Models (1996), Clarendon, Oxford
14. Read R., and Wilson R. An Atlas of Graphs (1998), Oxford Sci. Publ., Oxford
15. Semple C., and Steel M. Phylogenetics (2003), Oxford Univ. Press, Oxford
16. Speyer D., and Sturmfels B. The tropical Grassmannian. Adv. Geom. 4 (2004) 389-411
17. Spirtes P., Glymour C., and Scheines R. Causation, Prediction, and Search (2000), MIT Press, Cambridge, MA
18. Stanley R. Enumerative Combinatorics, vol. 2 (1999), Cambridge Univ. Press
19. Sturmfels B. Gröbner Bases and Convex Polytopes. Univ. Lecture Ser. vol. 8 (1996), Amer. Math. Soc., Providence, RI
20. Sturmfels B., and Sullivant S. Combinatorial secant varieties. Pure Appl. Math. Q. 2 (2006) 285-309