Logics for Approximating Implication Problems of Saturated Conditional Independence

  • Henning Koehler
  • Sebastian Link
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8761)


Random variables are declared complete whenever they must not admit missing data. Intuitively, the larger the set of complete random variables the closer the implication of saturated conditional independence statements is approximated. Two different notions of implication are studied. In the classical notion, a statement is implied jointly by a set of statements, the fixed set of random variables and its subset of complete random variables. For the notion of pure implication the set of random variables is left undetermined. A first axiomatization for the classical notion is established that distinguishes purely implied from classically implied statements. Axiomatic, algorithmic and logical characterizations of pure implication are established. The latter appeal to applications in which the existence of random variables is uncertain, for example, when statements are integrated from different sources, when random variables are unknown or when they shall remain hidden.


Approximation Conditional Independence Implication Missing Data Unknown Random Variable \(\mathcal{S}\)-3 Logics 


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  1. 1.
    Biskup, J.: Inferences of multivalued dependencies in fixed and undetermined universes. Theor. Comput. Sci. 10(1), 93–106 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Biskup, J., Link, S.: Appropriate inferences of data dependencies in relational databases. Ann. Math. Artif. Intell. 63(3-4), 213–255 (2012)MathSciNetCrossRefGoogle Scholar
  3. 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)CrossRefGoogle Scholar
  4. 4.
    Chickering, D.M., Heckerman, D.: Efficient approximations for the marginal likelihood of Bayesian networks with hidden variables. Machine Learning 29(2-3), 181–212 (1997)CrossRefzbMATHGoogle Scholar
  5. 5.
    Dawid, A.P.: Conditional independence in statistical theory. Journal of the Royal Statistical Society. Series B (Methodological) 41(1), 1–31 (1979)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Dempster, A., Laird, N.M., Rubin, D.: Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society B 39, 1–39 (1977)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Friedman, N.: Learning belief networks in the presence of missing values and hidden variables. In: Fisher, D.H. (ed.) Proceedings of the Fourteenth International Conference on Machine Learning (ICML 1997), Nashville, Tennessee, USA, July 8-12, pp. 125–133. Morgan Kaufmann (1997)Google Scholar
  8. 8.
    Galil, Z.: An almost linear-time algorithm for computing a dependency basis in a relational database. J. ACM 29(1), 96–102 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Geiger, D., Pearl, J.: Logical and algorithmic properties of conditional independence and graphical models. The Annals of Statistics 21(4), 2001–2021 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Halpern, J.Y.: Reasoning about uncertainty. MIT Press (2005)Google Scholar
  11. 11.
    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
  12. 12.
    Herrmann, C.: On the undecidability of implications between embedded multivalued database dependencies. Inf. Comput. 122(2), 221–235 (1995)CrossRefzbMATHGoogle Scholar
  13. 13.
    Koehler, H., Link, S.: Saturated conditional independence with fixed and undetermined sets of incomplete random variables. In: Zhang, N.L., Tian, J. (eds.) Proceedings of the Thirtieth Conference on Uncertainty in Artificial Intelligence, Quebec City, Quebec, Canada, July 23-27. AUAI Press (2013)Google Scholar
  14. 14.
    Koller, D., Friedman, N.: Probabilistic Graphical Models - Principles and Techniques. MIT Press (2009)Google Scholar
  15. 15.
    Lauritzen, S.: The EM algorithm for graphical association models with missing data. Computational Statistics and Data Analysis 19, 191–201 (1995)CrossRefzbMATHGoogle Scholar
  16. 16.
    Lenzerini, M., Schaerf, M.: The scientific legacy of Marco Cadoli in artificial intelligence. Intelligenza Artificiale 7(1), 1–5 (2013)Google Scholar
  17. 17.
    Lien, E.: On the equivalence of database models. J. ACM 29(2), 333–362 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Link, S.: Charting the completeness frontier of inference systems for multivalued dependencies. Acta Inf. 45(7-8), 565–591 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Link, S.: Characterizations of multivalued dependency implication over undetermined universes. J. Comput. Syst. Sci. 78(4), 1026–1044 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Link, S.: Sound approximate reasoning about saturated conditional probabilistic independence under controlled uncertainty. J. Applied Logic 11(3), 309–327 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Marlin, B.M., Zemel, R.S., Roweis, S.T., Slaney, M.: Recommender systems, missing data and statistical model estimation. In: Walsh, T. (ed.) Proceedings of the 22nd International Joint Conference on Artificial Intelligence, IJCAI 2011, Barcelona, Catalonia, Spain, July 16-22, pp. 2686–2691. IJCAI/AAAI (2011)Google Scholar
  22. 22.
    Niepert, M., Van Gucht, D., Gyssens, M.: Logical and algorithmic properties of stable conditional independence. Int. J. Approx. Reasoning 51(5), 531–543 (2010)CrossRefzbMATHGoogle Scholar
  23. 23.
    Niepert, M., Gyssens, M., Sayrafi, B., Gucht, D.V.: On the conditional independence implication problem: A lattice-theoretic approach. Artif. Intell. 202, 29–51 (2013)CrossRefGoogle Scholar
  24. 24.
    Pearl, J.: Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann, San Francisco (1988)Google Scholar
  25. 25.
    Saar-Tsechansky, M., Provost, F.J.: Handling missing values when applying classification models. Journal of Machine Learning Research 8, 1623–1657 (2007)zbMATHGoogle Scholar
  26. 26.
    Sagiv, Y., Delobel, C., Parker Jr., D.S., Fagin, R.: An equivalence between relational database dependencies and a fragment of propositional logic. J. ACM 28(3), 435–453 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Schaerf, M., Cadoli, M.: Tractable reasoning via approximation. Artif. Intell. 74, 249–310 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Singh, M.: Learning bayesian networks from incomplete data. In: Kuipers, B., Webber, B.L. (eds.) Proceedings of the Fourteenth National Conference on Artificial Intelligence and Ninth Innovative Applications of Artificial Intelligence Conference, AAAI 1997, IAAI 1997, Providence, Rhode Island, July 27-31, pp. 534–539. AAAI Press/The MIT Press (1997)Google Scholar
  29. 29.
    Stott Parker Jr., D., Parsaye-Ghomi, K.: Inferences involving embedded multivalued dependencies and transitive dependencies. In: Chen, P.P., Sprowls, R.C. (eds.) Proceedings of the 1980 ACM SIGMOD International Conference on Management of Data, Santa Monica, California, May 14-16, pp. 52–57. ACM Press (1980)Google Scholar
  30. 30.
    Studený, M.: Conditional independence relations have no finite complete characterization. In: Ámos Víšek, J. (ed.) Transactions of the 11th Prague Conference on Information Theory, Statistical Decision Functions and Random Processes, Prague, Czech Republic, August 27-31, 1990, pp. 377–396. Academia (1992)Google Scholar
  31. 31.
    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)CrossRefGoogle Scholar
  32. 32.
    Zhu, X., Zhang, S., Zhang, J., Zhang, C.: Cost-sensitive imputing missing values with ordering. In: Proceedings of the Twenty-Second AAAI Conference on Artificial Intelligence, Vancouver, British Columbia, Canada, July 22-26, pp. 1922–1923. AAAI Press (2007)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Henning Koehler
    • 1
  • Sebastian Link
    • 2
  1. 1.School of Engineering & Advanced TechnologyMassey UniversityNew Zealand
  2. 2.Department of Computer ScienceUniversity of AucklandNew Zealand

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