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Conditional Independences in Gaussian Vectors and Rings of Polynomials

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Conditionals, Information, and Inference (WCII 2002)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 3301))

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Abstract

Inference among the conditional independences in nondegenerate Gaussian vectors is studied by algebraic techniques. A general method to prove implications involving the conditional independences is presented. The method relies on computations of a Groebner basis. Examples of the implications are discussed.

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References

  1. Csirmaz, L.: The dealer’s random bits in perfect secret-sharing schemes. Studia Scient. Math. Hungarica 32, 429–437 (1996)

    MathSciNet  MATH  Google Scholar 

  2. Cowell, R.G., Dawid, A.P., Lauritzen, S.L., Spiegelhalter, D.J.: Probabilistic Networks and Expert Systems. Springer, New York (1999)

    MATH  Google Scholar 

  3. Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms. In: Undergraduate Texts in Mathematics. Springer, New York (1997)

    Google Scholar 

  4. DeConcini, C., Eisenbud, D., Procesi, D.: Young diagrams and determinantal varieties. Inventiones math. 56, 129–165 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  5. Doubilet, P., Rota, G.C., Stein, J.: On the foundations of the combinatorial theory IX: combinatorial methods in invariant theory. Studies in Applied Math. 53, 185–216 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  6. Geiger, D., Meek, C.: Quantifier Elimination for statistical problems. In: Laskey, K., Prade, H. (eds.) Proceedings of Fifteenth Conference on Uncertainty in Artificial Intelligence, pp. 226–235. Morgan Kaufmann, San Francisco (1999)

    Google Scholar 

  7. Lauritzen, S.L.: Graphical Models. Oxford University Press, Oxford (1996)

    MATH  Google Scholar 

  8. Matúš, F.: On equivalence of Markov properties over undirected graphs. Journal of Applied Probability 29, 745–749 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  9. Matúš, F.: Conditional independence structures examined via minors. Annals of Mathematics and Artificial Intelligence 21, 99–128 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  10. Matúš, F.: Conditional independences among four random variables III: final conclusion. Combinatorics, Probability and Computing 8, 269–276 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  11. Matúš, F.: Matroid representations by partitions. Discrete Mathematics 203, 169–194 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  12. Mishra, B.: Algorithmic Algebra. In: Texts and monographs in Computer Science, vol. XIV. Springer, New York (1993)

    Google Scholar 

  13. Mond, D., Pellikaan, R.: Fitting ideals and multiple points of analytic mappings. In: de Arellano, E.R. (ed.) Algebraic and Analytic Geometry, Patzcuaro 1987. Springer Lecture Notes in Mathematics, vol. 1414, pp. 107–161 (1989)

    Google Scholar 

  14. Pearl, J.: Probabilistic Reasoning in Intelligent Systems. Morgan Kaufman, San Meteo (1988)

    MATH  Google Scholar 

  15. Prasolov, V.V.: Problems and Theorems in Linear Algebra. In: AMS Translations of Mathematical Monographs, vol. 134. AMS, Providence (1994)

    Google Scholar 

  16. Sagiv, Y., Walecka, S.F.: Subset dependencies and a completeness result for a subclass of embedded multivalued dependencies. Journal of ACM 29, 103–117 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  17. Spencer, J.: Ten Lectures on the Probabilistic Method. SIAM, Philadelphia (1994)

    Book  MATH  Google Scholar 

  18. Studený, M.: Conditional independence relations have no finite complete characterization. In: Transactions of the 11-th Prague Conference on Information Theory, Statistical Decision Functions and Random Processes, Academia, Prague, vol. B, pp. 377–396. Kluwer, Dordrecht (1991)

    Google Scholar 

  19. Studený, M.: On stochastic conditional independence: the problems of characterization and description. Annals of Mathematics and Artificial Intelligence 35, 323–341 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  20. Yeung, R.W.: A First Course in Information Theory. Kluwer Academic/Plenum Publishers (2002)

    Google Scholar 

  21. Zhang, Z., Yeung, R.W.: A non-Shannon-type conditional inequality of information quantities. IEEE Trans. Information Theory 43, 1982–1986 (1997)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Institute of Information Theory and Automation, Pod vodárenskou věží 4, 182 08, Prague, Czech Republic

    František Matúš

Authors
  1. František Matúš
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Editor information

Editors and Affiliations

  1. Dept. of Computer Science, TU Dortmund, 44221, Dortmund, Germany

    Gabriele Kern-Isberner

  2. FernUniversität in Hagen, Fachbereich Wirtschaftswissenschaft, Lehrstuhl für BWL, insb. Operations Research, 58084, Hagen, Germany

    Wilhelm Rödder

  3. FernUniversität in Hagen, Fachbereich Wirtschaftswissenschaft, Lehrstuhl BWL, insb. Operations Research, 58084, Hagen, Germany

    Friedhelm Kulmann

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Matúš, F. (2005). Conditional Independences in Gaussian Vectors and Rings of Polynomials. In: Kern-Isberner, G., Rödder, W., Kulmann, F. (eds) Conditionals, Information, and Inference. WCII 2002. Lecture Notes in Computer Science(), vol 3301. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11408017_9

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  • DOI: https://doi.org/10.1007/11408017_9

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-25332-7

  • Online ISBN: 978-3-540-32235-1

  • eBook Packages: Computer ScienceComputer Science (R0)

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