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Keywords:
multivariate Gaussian distribution; positive definite matrices; determinants; principal minors; conditional independence; probabilistic representability; semigraphoids; separation graphoids; gaussoids; covariance selection models; Markov perfectness
multivariate Gaussian distribution; positive definite matrices; determinants; principal minors; conditional independence; probabilistic representability; semigraphoids; separation graphoids; gaussoids; covariance selection models; Markov perfectness
Summary:
The simultaneous occurrence of conditional independences among subvectors of a regular Gaussian vector is examined. All configurations of the conditional independences within four jointly regular Gaussian variables are found and completely characterized in terms of implications involving conditional independence statements. The statements induced by the separation in any simple graph are shown to correspond to such a configuration within a regular Gaussian vector.
References:
[1] Cox D., Little, J., O’Shea D.: Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra. Second Edition. Springer-Verlag, New York 1997 MR 1417938 | Zbl 1118.13001
[2] Dawid A. P.: Conditional independence in statistical theory. J. Roy. Statist. Soc. Ser. B 41 (1979), 1–31 MR 0535541 | Zbl 0408.62004
[3] Dawid A. P.: Conditional independence for statistical operations. Ann. Statist. 8 (1980), 598–617 MR 0568723 | Zbl 0434.62006
[4] Dempster A. P.: Covariance selection. Biometrics 28 (1972), 157–175
[5] Frydenberg M.: Marginalization and collapsibility in graphical interaction models. Ann. Statist. 18 (1990), 790–805 MR 1056337 | Zbl 0725.62057
[7] Kauermann G.: On a dualization of graphical Gaussian models. Scand. J. Statist. 23 (1996), 105–116 MR 1380485 | Zbl 0912.62006
[8] Kurosh A.: Higher Algebra. Mir Publishers, Moscow 1975 MR 0384363 | Zbl 0546.00002
[9] Lauritzen S. L.: Graphical Models. (Oxford Statistical Science Series 17.) Oxford University Press, New York 1996 MR 1419991 | Zbl 1055.62126
[10] Levitz M., Perlman M. D., Madigan D.: Separation and completeness properties for AMP chain graph Markov models. Ann. Statist. 29 (2001), 1751–1784 MR 1891745 | Zbl 1043.62080
[11] Lněnička R.: 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] Matúš F.: 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] Matúš F.: On equivalence of Markov properties over undirected graphs. J. Appl. Probab. 29 (1992), 745–749 MR 1174448 | Zbl 0753.68080
[14] Matúš F.: Conditional independences among four random variables II. Combin. Probab. Comput. 4 (1995), 407–417 MR 1377558 | Zbl 0846.60004
[15] Matúš F.: Conditional independence structures examined via minors. Ann. Math. Artif. Intell. 21 (1997), 99–128 MR 1479010 | Zbl 0888.68097
[16] Matúš F.: Conditional independences among four random variables III: final conclusion. Combin. Probab. Comput. 8 (1999), 269–276 MR 1702569 | Zbl 0941.60004
[17] Matúš F.: 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 Zbl 1111.68685
[18] Matúš F.: Towards classification of semigraphoids. Discrete Math. 277 (2004), 115–145 MR 2033729 | Zbl 1059.68082
[19] Matúš F., Studený M.: Conditional independences among four random variables I. Combin. Probab. Comput. 4 (1995), 269–278 Zbl 0839.60004
[20] Pearl J.: Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann, San Mateo, Calif. 1988 MR 0965765 | Zbl 0746.68089
[21] Prasolov V. V.: Problems and Theorems in Linear Algebra. (Translations of Mathematical Monographs 134.) American Mathematical Society 1996 MR 1277174 | Zbl 1185.15001
[22] Studený M.: 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] Studený M.: Structural semigraphoids. Internat. J. Gen. Syst. 22 (1994), 207–217 Zbl 0797.60006
[24] Studený M.: Probabilistic Conditional Independence Structures. Springer, London 2005
[26] Šimeček P.: 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] Šimeček P.: 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] Whittaker J.: Graphical Models in Applied Multivariate Statistics. Wiley, New York 1990 MR 1112133 | Zbl 1151.62053
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