On Gaussian conditional independence structures

Kybernetika

Volumn 43, Issue 3, 2007, Pages 327-342

  Lněnička, Radim ^a   Matúš, František ^a  

^a INSTITUTE OF INFORMATION THEORY AND AUTOMATION   (Czech Republic)

Author keywords

Conditional independence; Covariance selection models; Determinants; Gaussoids; Markov perfectness; Multivariate Gaussian distribution; Positive definite matrices; Principal minors; Probabilistic representability; Semi graphoids; Separation graphoids

Indexed keywords

EID: 38349195395     PISSN: 00235954     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (28)

1. D. Cox, J. Little, and D. O'Shea: Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra. Second Edition. Springer-Verlag, New York 1997.
2. A. P. Dawid: Conditional independence in statistical theory. J. Roy. Statist. Soe. Ser. B 41 (1979), 1-31.
3. A. P. Dawid: Conditional independence for statistical operations. Ann. Statist. 8 (1980), 598-617.
4. A. P. Dempster: Covariance selection. Biometrics 28 (1972), 157-175.
5. M. Frydenberg: Marginalization and collapsibility in graphical interaction models. Ann. Statist. 18 (1990), 790-805.
6. O. Holtz and B. Sturmfels: Hyperdeterminantal relations among symmetric principal minors. (unpublished manuscript)
7. G. Kauermann: On a dualization of graphical Gaussian models. Scand. J. Statist. 23 (1996), 105-116.
8. A. Kurosh: Higher Algebra. Mir Publishers, Moscow 1975.
9. S. L. Lauritzen: Graphical Models. (Oxford Statistical Science Series 17.) Oxford University Press, New York 1996.
10. M. Levitz, M. D. Perlman, and D. Madigan: Separation and completeness properties for AMP chain graph Markov models. Ann. Statist. 29 (2001), 1751-1784.
11. R. Lněnička: On conditional independences among four Gaussian variables. In: Proc. Conditionals, Information, and Inference - WCII'04 (G. Kern-Isberner, W. Roedder, and F. Kulmann, eds.), Universität Ulm, Ulm 2004, pp. 89-101.
12. F. Matúš: Ascending and descending conditional independence relations. In: Trans. 11th Prague Conference on Information Theory, Statistical Decision Functions and Random Processes (S. Kubík and J. Á. Víšek, eds.), Vol. B, Academia, Prague, and Kluwer, Dordrecht 1991, pp. 189-200.
13. F. Matúš: On equivalence of Markov properties over undirected graphs. J. Appl. Probab. 29 (1992), 745-749.
14. F. Matúš: Conditional independences among four random variables II. Combin. Probab. Comput. 4 (1995), 407-417.
15. F. Matúš: Conditional independence structures examined via minors. Ann. Math. Artif. Intell. 21 (1997), 99-128.
16. F. Matúš: Conditional independences among four random variables III: final conclusion. Combin. Probab. Comput. 8 (1999), 269-276.
17. F. Matúš: Conditional independences in Gaussian vectors and rings of polynomials. In: Proc. Conditionals, Information, and Inference - WCII 2002 (Lecture Notes in Computer Science 3301; G. Kern-Isberner, W. Rödder, and F. Kulmann, eds.), Springer, Berlin 2005, pp. 152-161.
18. F. Matúš: Towards classification of semigraphoids. Discrete Math. 277 (2004), 115-145.
19. F. Matúš and M. Studený: Conditional independences among four random variables I. Combin. Probab. Comput. 4 (1995), 269-278.
20. J. Pearl: Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann, San Mateo, Calif. 1988.
21. V. V. Prasolov: Problems and Theorems in Linear Algebra. (Translations of Mathematical Monographs 134.) American Mathematical Society 1996.
22. M. Studený: Conditional independence relations have no finite complete characterization. In: Trans. 11th Prague Conference on Information Theory, Statistical Decision Functions and Random Processes (S. Kubík and J. Á. Víšek, eds.), Vol. B, Academia, Prague, and Kluwer, Dordrecht 1991, pp. 377-396.
23. M. Studený: Structural semigraphoids. Internat. J. Gen. Syst. 22 (1994), 207-217.
24. M. Studený: Probabilistic Conditional Independence Structures. Springer, London 2005.
25. S. Sullivant: Gaussian conditional independence has no finite complete axiom system. (unpublished manuscript)
26. P. Šimeček: Classes of Gaussian, discrete and binary representable independence models have no finite characterization. In: Prague Stochastics (M. Hušková and M. Janžura, eds.), Matfyzpress, Charles University, Prague 2006, pp. 622-632.
27. P. Šimeček: Gaussian representation of independence models over four random variables. In: Proc. COMPSTAT 2006 - World Conference on Computational Statistics 17 (A. Rizzi and M. Vichi, eds.), Rome 2006, pp. 1405-1412.
28. J. Whittaker: Graphical Models in Applied Multivariate Statistics. Wiley, New York 1990.