On Gaussian conditional independence structures

Radim Lněnička; František Matúš

Kybernetika (2007)

  • Volume: 43, Issue: 3, page 327-342
  • ISSN: 0023-5954

Abstract

top
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.

How to cite

top

Lněnička, Radim, and Matúš, František. "On Gaussian conditional independence structures." Kybernetika 43.3 (2007): 327-342. <http://eudml.org/doc/33861>.

@article{Lněnička2007,
abstract = {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.},
author = {Lněnička, Radim, Matúš, František},
journal = {Kybernetika},
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; separation graphoids; gaussoids; covariance selection models; Markov perfectness},
language = {eng},
number = {3},
pages = {327-342},
publisher = {Institute of Information Theory and Automation AS CR},
title = {On Gaussian conditional independence structures},
url = {http://eudml.org/doc/33861},
volume = {43},
year = {2007},
}

TY - JOUR
AU - Lněnička, Radim
AU - Matúš, František
TI - On Gaussian conditional independence structures
JO - Kybernetika
PY - 2007
PB - Institute of Information Theory and Automation AS CR
VL - 43
IS - 3
SP - 327
EP - 342
AB - 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.
LA - eng
KW - 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; separation graphoids; gaussoids; covariance selection models; Markov perfectness
UR - http://eudml.org/doc/33861
ER -

References

top
  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 Zbl1118.13001MR1417938
  2. Dawid A. P., Conditional independence in statistical theory, J. Roy. Statist. Soc. Ser. B 41 (1979), 1–31 (1979) Zbl0408.62004MR0535541
  3. Dawid A. P., Conditional independence for statistical operations, Ann. Statist. 8 (1980), 598–617 (1980) Zbl0434.62006MR0568723
  4. Dempster A. P., Covariance selection, Biometrics 28 (1972), 157–175 (1972) 
  5. Frydenberg M., Marginalization and collapsibility in graphical interaction models, Ann. Statist. 18 (1990), 790–805 (1990) Zbl0725.62057MR1056337
  6. Kauermann G., On a dualization of graphical Gaussian models, Scand. J. Statist. 23 (1996), 105–116 (1996) Zbl0912.62006MR1380485
  7. Kurosh A., Higher Algebra, Mir Publishers, Moscow 1975 Zbl0546.00002MR0384363
  8. Lauritzen S. L., Graphical Models, (Oxford Statistical Science Series 17.) Oxford University Press, New York 1996 Zbl1055.62126MR1419991
  9. Levitz M., Perlman M. D., Madigan D., Separation and completeness properties for AMP chain graph Markov models, Ann. Statist. 29 (2001), 1751–1784 Zbl1043.62080MR1891745
  10. 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 
  11. 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 (1991) 
  12. Matúš F., On equivalence of Markov properties over undirected graphs, J. Appl. Probab. 29 (1992), 745–749 (1992) Zbl0753.68080MR1174448
  13. Matúš F., Conditional independences among four random variables II, Combin. Probab. Comput. 4 (1995), 407–417 (1995) Zbl0846.60004MR1377558
  14. Matúš F., Conditional independence structures examined via minors, Ann. Math. Artif. Intell. 21 (1997), 99–128 (1997) Zbl0888.68097MR1479010
  15. Matúš F., Conditional independences among four random variables III: final conclusion, Combin. Probab. Comput. 8 (1999), 269–276 (1999) Zbl0941.60004MR1702569
  16. 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 Zbl1111.68685
  17. Matúš F., Towards classification of semigraphoids, Discrete Math. 277 (2004), 115–145 Zbl1059.68082MR2033729
  18. Matúš F., Studený M., Conditional independences among four random variables I, Combin. Probab. Comput. 4 (1995), 269–278 (1995) Zbl0839.60004
  19. Pearl J., Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference, Morgan Kaufmann, San Mateo, Calif. 1988 Zbl0746.68089MR0965765
  20. Prasolov V. V., Problems and Theorems in Linear Algebra, (Translations of Mathematical Monographs 134.) American Mathematical Society 1996 Zbl1185.15001MR1277174
  21. 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 (1991) 
  22. Studený M., Structural semigraphoids, Internat. J. Gen. Syst. 22 (1994), 207–217 (1994) Zbl0797.60006
  23. Studený M., Probabilistic Conditional Independence Structures, Springer, London 2005 
  24. 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 
  25. 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 
  26. Whittaker J., Graphical Models in Applied Multivariate Statistics, Wiley, New York 1990 Zbl1151.62053MR1112133

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.