Skip to main content
SpringerLink
Log in

Approximate Reasoning about Generalized Conditional Independence with Complete Random Variables

  • Conference paper
Scalable Uncertainty Management (SUM 2013)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 8078))

Included in the following conference series:

  • 622 Accesses

Abstract

The implication problem of conditional statements about the independence of finitely many sets of random variables is studied in the presence of controlled uncertainty. Uncertainty refers to the possibility of missing data. As a control mechanism random variables can be declared complete, in which case data on these random variables cannot be missing. While the implication of conditional independence statements is not axiomatizable, a finite Horn axiomatization is established for the expressive class of saturated conditional independence statements under controlled uncertainty. Complete random variables allow us to balance the expressivity of sets of saturated statements with the efficiency of deciding their implication. This ability can soundly approximate reasoning in the absence of missing data. Delobel’s class of full first-order hierarchical database decompositions are generalized to the presence of controlled uncertainty, and their implication problem shown to be equivalent to that of saturated conditional independence.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Atzeni, P., Morfuni, N.: Functional dependencies and constraints on null values in database relations. Information and Control 70(1), 1–31 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  2. Batista, G., Monard, M.: An analysis of four missing data treatment methods for supervised learning. Applied Artificial Intelligence 17(5-6), 519–533 (2003)

    Article  Google Scholar 

  3. Biskup, J., Hartmann, S., Link, S.: Probabilistic conditional independence under schema certainty and uncertainty. In: Hüllermeier, E., Link, S., Fober, T., Seeger, B. (eds.) SUM 2012. LNCS, vol. 7520, pp. 365–378. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  4. Chickering, D., Heckerman, D.: Efficient approximations for the marginal likelihood of Bayesian networks with hidden variables. Machine Learning 29(2-3), 181–212 (1997)

    Article  MATH  Google Scholar 

  5. Darwiche, A.: Modeling and Reasoning with Bayesian Networks. Cambridge University Press (2009)

    Google Scholar 

  6. Dawid, A.: Conditional independence in statistical theory. Journal of the Royal Statistical Society 41(1), 1–31 (1979)

    MathSciNet  MATH  Google Scholar 

  7. Delobel, C.: Normalization and hierarchical dependencies in the relational data model. ACM Trans. Database Syst. 3(3), 201–222 (1978)

    Article  Google Scholar 

  8. Dempster, A., Laird, N., Rubin, D.: Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society B 39, 139 (1977)

    MathSciNet  Google Scholar 

  9. Fayyad, U., Piatetsky-Shapiro, G., Smyth, P.: From data mining to knowledge discovery in databases. AI Magazine 17(3), 37–54 (1996)

    Google Scholar 

  10. Ferrarotti, F., Hartmann, S., Link, S.: Reasoning about functional and full hierarchical dependencies over partial relations. Inf. Sci. 235, 150–173 (2013)

    Article  MathSciNet  Google Scholar 

  11. Friedman, N.: Learning belief networks in the presence of missing values and hidden variables. In: ICML, pp. 125–133 (1997)

    Google Scholar 

  12. Galil, Z.: An almost linear-time algorithm for computing a dependency basis in a relational database. J. ACM 29(1), 96–102 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  13. Geiger, D., Pearl, J.: Logical and algorithmic properties of conditional independence and graphical models. The Annals of Statistics 21(4), 2001–2021 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  14. Geiger, D., Pearl, J.: Logical and algorithmic properties of independence and their application to Bayesian networks. Ann. Math. Artif. Intell. 2, 165–178 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  15. Grädel, E., Väänänen, J.: Dependence and independence. Studia Logica 101(2), 399–410 (2012)

    Article  Google Scholar 

  16. Graetzer, G.: General lattice theory. Birkhäuser, Boston (1998)

    MATH  Google Scholar 

  17. Halpern, J.: Reasoning about uncertainty. MIT Press (2005)

    Google Scholar 

  18. Hartmann, S., Link, S.: The implication problem of data dependencies over SQL table definitions: axiomatic, algorithmic and logical characterizations. ACM Trans. Database Syst. 37(2), Article 13 (2012)

    Google Scholar 

  19. Herrmann, C.: Corrigendum to “On the undecidability of implications between embedded multivalued database dependencies”. Inf. Comput. 204(12), 1847–1851 (2006)

    Article  MATH  Google Scholar 

  20. Kontinen, J., Link, S., Väänänen, J.: Independence in database relations. In: Libkin, L. (ed.) WoLLIC 2013. LNCS, vol. 8071, pp. 179–193. Springer, Heidelberg (2013)

    Google Scholar 

  21. Lauritzen, S.: The EM algorithm for graphical association models with missing data. Computational Statistics and Data Analysis 19, 191–201 (1995)

    Google Scholar 

  22. Lien, E.: On the equivalence of database models. J. ACM 29(2), 333–362 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  23. Link, S.: Propositional reasoning about saturated conditional probabilistic independence. In: Ong, L., de Queiroz, R. (eds.) WoLLIC 2012. LNCS, vol. 7456, pp. 257–267. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  24. Link, S.: Reasoning about saturated conditional independence under uncertainty: Axioms, algorithms, and Levesque’s situations to the rescue. AAAI (2013)

    Google Scholar 

  25. Link, S.: Sound approximate reasoning about saturated probabilistic conditional independence under controlled uncertainty. J. Applied Logic (2013), http://dx.doi.org/10.1016/j.jal.2013.05.004

  26. Marlin, B., Zemel, R., Roweis, S., Slaney, M.: Recommender systems, missing data and statistical model estimation. In: IJCAI, pp. 2686–2691 (2011)

    Google Scholar 

  27. Matúš, F.: Ascending and descending conditional independence relations. In: Transactions of the 11th Prague Conference on Information Theory, Statistical Decision Functions and Random Processes, pp. 189–200 (1992)

    Google Scholar 

  28. Niepert, M., Van Gucht, D., Gyssens, M.: Logical and algorithmic properties of stable conditional independence. Int. J. Approx. Reasoning 51(5), 531–543 (2010)

    Article  MATH  Google Scholar 

  29. Niepert, M., Van Gucht, D., Gyssens, M.: On the conditional independence implication problem: A lattice-theoretic approach. In: UAI, pp. 435–443 (2008)

    Google Scholar 

  30. Pearl, J.: Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann, San Francisco (1988)

    Google Scholar 

  31. Singh, M.: Learning Bayesian networks from incomplete data. In: AAAI, pp. 534–539 (1997)

    Google Scholar 

  32. Stott Parker Jr., D., Parsaye-Ghomi, K.: Inferences involving embedded multivalued dependencies and transitive dependencies. In: SIGMOD, pp. 52–57 (1980)

    Google Scholar 

  33. Studený, M.: Conditional independence relations have no finite complete characterization. In: Transactions of the 11th Prague Conference on Information Theory, Statistical Decision Functions and Random Processes, pp. 377–396 (1992)

    Google Scholar 

  34. Thiesson, B.: Accelerated quantification of bayesian networks with incomplete data. In: KDD, pp. 306–311 (1995)

    Google Scholar 

  35. de Waal, P., van der Gaag, L.: Stable independence in perfect maps. In: UAI, pp. 161–168 (2005)

    Google Scholar 

  36. Wong, S., Butz, C., Wu, D.: On the implication problem for probabilistic conditional independency. IEEE Trans. Systems, Man, and Cybernetics, Part A: Systems and Humans 30(6), 785–805 (2000)

    Article  Google Scholar 

  37. Zaniolo, C.: Database relations with null values. J. Comput. Syst. Sci. 28(1), 142–166 (1984)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, The University of Auckland, New Zealand

    Sebastian Link

Authors

Editor information

Editors and Affiliations

  1. School of Electronics, Electrical Engineering and Computer Science, Queen’s University Belfast, BT9 5BN, Belfast, UK

    Weiru Liu

  2. Department of Computer Science, University of Maryland, 20742, College Park, MD, USA

    V. S. Subrahmanian

  3. Département d’Informatique, Université de Mons, 7000, Mons, Belgium

    Jef Wijsen

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Link, S. (2013). Approximate Reasoning about Generalized Conditional Independence with Complete Random Variables. In: Liu, W., Subrahmanian, V.S., Wijsen, J. (eds) Scalable Uncertainty Management. SUM 2013. Lecture Notes in Computer Science(), vol 8078. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40381-1_21

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-40381-1_21

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-40380-4

  • Online ISBN: 978-3-642-40381-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation