A Logical Approach to Context-Specific Independence

  • Jukka Corander
  • Antti Hyttinen
  • Juha Kontinen
  • Johan Pensar
  • Jouko Väänänen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9803)

Abstract

Bayesian networks constitute a qualitative representation for conditional independence (CI) properties of a probability distribution. It is known that every CI statement implied by the topology of a Bayesian network G is witnessed over G under a graph-theoretic criterion called d-separation. Alternatively, all such implied CI statements have been shown to be derivable using the so-called semi-graphoid axioms. In this article we consider Labeled Directed Acyclic Graphs (LDAG) the purpose of which is to graphically model situations exhibiting context-specific independence (CSI). We define an analogue of dependence logic suitable to express context-specific independence and study its basic properties. We also consider the problem of finding inference rules for deriving non-local CSI and CI statements that logically follow from the structure of a LDAG but are not explicitly encoded by it.

Notes

Acknowledgements

The third author was supported by grant 292767 of the Academy of Finland. The fourth author was supported by FDPSS via grant 141318 of the Academy of Finland.

References

  1. 1.
    Boutilier, C., Friedman, N., Goldszmidt, M., Koller, D.: Context-specific independence in Bayesian networks. In: Proceedings of the Twelfth International Conference on Uncertainty in Artificial Intelligence. UAI 1996, pp. 115–123. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA (1996). http://dl.acm.org/citation.cfm?id=2074284.2074298
  2. 2.
    Dawid, A.P.: Conditional independence in statistical theory. J. Roy. Stat. Soc. Ser. B (Methodological) 41(1), 1–31 (1979). doi: 10.2307/2984718 MathSciNetMATHGoogle Scholar
  3. 3.
    Durand, A., Hannula, M., Kontinen, J., Meier, A., Virtema, J.: Approximation and dependence via multiteam semantics. In: Gyssens, M., et al. (eds.) FoIKS 2016. LNCS, vol. 9616, pp. 271–291. Springer, Heidelberg (2016). doi: 10.1007/978-3-319-30024-5_15 CrossRefGoogle Scholar
  4. 4.
    Galliani, P.: Inclusion and exclusion dependencies in team semantics - on some logics of imperfect information. Ann. Pure Appl. Logic 163(1), 68–84 (2012)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Geiger, D., Paz, A., Pearl, J.: Axioms and algorithms for inferences involving probabilistic independence. Inf. Comput. 91(1), 128–141 (1991)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Geiger, D., Pearl, J.: On the logic of causal models. In: Proceedings of the Fourth Annual Conference on Uncertainty in Artificial Intelligence. UAI 1988, pp. 3–14. North-Holland Publishing Co., Amsterdam, The Netherlands (1990). http://dl.acm.org/citation.cfm?id=647231.719429
  7. 7.
    Geiger, D., Verma, T., Pearl, J.: Identifying independence in Bayesian networks. Networks 20(5), 507–534 (1990)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Grädel, E., Väänänen, J.A.: Dependence and independence. Stud. Logica 101(2), 399–410 (2013)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Gyssens, M., Niepert, M., Gucht, D.V.: On the completeness of the semigraphoid axioms for deriving arbitrary from saturated conditional independence statements. Inf. Process. Lett. 114(11), 628–633 (2014). http://www.sciencedirect.com/science/article/pii/S0020019014001057 MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Hannula, M.: Axiomatizing first-order consequences in independence logic. Ann. Pure Appl. Logic 166(1), 61–91 (2015). doi: 10.1016/j.apal.2014.09.002 MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Hannula, M.: Reasoning about embedded dependencies using inclusion dependencies. In: Davis, M., Fehnker, A., McIver, A., Voronkov, A. (eds.) LPAR-20 2015. LNCS, vol. 9450, pp. 16–30. Springer, Heidelberg (2015). doi: 10.1007/978-3-662-48899-7_2 CrossRefGoogle Scholar
  12. 12.
    Hannula, M., Kontinen, J.: A finite axiomatization of conditional independence and inclusion dependencies. In: Beierle, C., Meghini, C. (eds.) FoIKS 2014. LNCS, vol. 8367, pp. 211–229. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  13. 13.
    Herrmann, C.: On the undecidability of implications between embedded multivalued database dependencies. Inf. Comput. 122(2), 221–235 (1995)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Koller, D., Friedman, N.: Probabilistic graphical models: principles and techniques. MIT Press, Cambridge (2009)MATHGoogle Scholar
  15. 15.
    Kontinen, J., Väänänen, J.A.: Axiomatizing first-order consequences in dependence logic. Ann. Pure Appl. Logic 164(11), 1101–1117 (2013)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Link, S.: Reasoning about saturated conditional independence under uncertainty: axioms, algorithms, and levesque’s situations to the rescue. In: Proceedings of AAAI. AAAI Press (2013)Google Scholar
  17. 17.
    Link, S.: Sound approximate reasoning about saturated conditional probabilistic independence under controlled uncertainty. J. Appl. Logic 11(3), 309–327 (2013)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Link, S.: Frontiers for propositional reasoning about fragments of probabilistic conditional independence and hierarchical database decompositions. Theor. Comput. Sci. 603, 111–131 (2015)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    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). doi: 10.1016/j.artint.2013.06.005 MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Nyman, H., Pensar, J., Corander, J.: Context-specific and local independence in Markovian dependence structures. In: Dependence Logic: Theory and Applications. Springer (To appear) (2016)Google Scholar
  21. 21.
    Pearl, J.: Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann, San Francisco (1988)MATHGoogle Scholar
  22. 22.
    Pensar, J., Nyman, H.J., Koski, T., Corander, J.: Labeled directed acyclic graphs: a generalization of context-specific independence in directed graphical models. Data Min. Knowl. Discov. 29(2), 503–533 (2015). doi: 10.1007/s10618-014-0355-0 MathSciNetCrossRefGoogle Scholar
  23. 23.
    Studeny, M.: Conditional independence relations have no finite complete characterization. In: Kubik, S., Visek, J. (eds.) Transactions of the 11th Prague Conference. Information Theory, Statistical Decision Functions and Random Processes, vol. B, pp. 377–396. Kluwer, Dordrecht (1992)Google Scholar
  24. 24.
    Väänänen, J.: Dependence logic: A New Approach to Independence Friendly Logic, London Mathematical Society Student Texts, vol. 70. Cambridge University Press, Cambridge (2007)CrossRefMATHGoogle Scholar
  25. 25.
    Verma, T., Pearl, J.: Causal networks: semantics and expressiveness. In: Shachter, R.D., Levitt, T.S., Kanal, L.N., Lemmer, J.F. (eds.) Proceedings of the Fourth Annual Conference on Uncertainty in Artificial Intelligence, Minneapolis, MN, USA, 10–12 July 1988. UAI 1988, pp. 69–78. North-Holland (1988)Google Scholar
  26. 26.
    Wong, S., Butz, C., Wu, D.: On the implication problem for probabilistic conditional independency. IEEE Trans. Syst. Man Cybern. Part A: Syst. Hum. 30(6), 785–805 (2000)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Jukka Corander
    • 1
    • 2
  • Antti Hyttinen
    • 3
  • Juha Kontinen
    • 1
  • Johan Pensar
    • 4
  • Jouko Väänänen
    • 1
    • 5
  1. 1.Department of Mathematics and StatisticsUniversity of HelsinkiHelsinkiFinland
  2. 2.Department of BiostatisticsUniversity of OsloOsloNorway
  3. 3.HIIT, Department of Computer ScienceUniversity of HelsinkiHelsinkiFinland
  4. 4.Department of Mathematics and StatisticsÅbo Akademi UniversityTurkuFinland
  5. 5.Institute for Logic, Language and ComputationUniversity of AmsterdamAmsterdamThe Netherlands

Personalised recommendations