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DL-Lite Bayesian Networks: A Tractable Probabilistic Graphical Model

  • Denis D. MauáEmail author
  • Fabio G. Cozman
Conference paper
  • 411 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9310)

Abstract

The construction of probabilistic models that can represent large systems requires the ability to describe repetitive and hierarchical structures. To do so, one can resort to constructs from description logics. In this paper we present a class of relational Bayesian networks based on the popular description logic DL-Lite. Our main result is that, for this modeling language, marginal inference and most probable explanation require polynomial effort. We show this by reductions to edge covering problems, and derive a result of independent interest; namely, that counting edge covers in a particular class of graphs requires polynomial effort.

Notes

Acknowledgements

The second author was partially supported by CNPq.

References

  1. 1.
    Artale, A., Calvanese, D., Kontchakov, R., Zakharyashev, M.: The DL-Lite family and relations. J. Artif. Intell. Res. 36, 1–69 (2009)zbMATHGoogle Scholar
  2. 2.
    Baader, F., Nutt, W.: Basic description logics. In: Baader, F., McGuinness, D.L., Nardi, D., Patel-Schneider, P.F. (eds.) Description Logic Handbook, pp. 47–100. Cambridge University Press, Cambridge (2002)Google Scholar
  3. 3.
    Bacchus, F.: Representing and Reasoning with Probabilistic Knowledge: A Logical Approach. MIT Press, Cambridge (1990)Google Scholar
  4. 4.
    van den Broeck, G.: On the completeness of first-order knowledge compilation for lifted probabilistic inference. In: Advances in Neural Processing Information Systems (2011)Google Scholar
  5. 5.
    van den Broeck, G., Wannes, M., Darwiche, A.: Skolemization for weighted first-order model counting. In: Proceedings of the International Conference on Principles of Knowledge Representation and Reasoning (2014)Google Scholar
  6. 6.
    Cai, J.Y., Lu, P., Xia, M.: Holographic reduction, interpolation and hardness. Comput. Complex. 21(4), 573–604 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Calvanese, D., De Giacomo, G., Lembo, D. abd Lenzerini, M., Rosati, R.: DL-Lite: tractable description logics for ontologies. In: Proceedings of the AAAI Conference, pp. 602–607 (2005)Google Scholar
  8. 8.
    Ceylan, I., Peñaloza, R.: The Bayesian description logic \(\lfloor \rceil \! \updownarrow \) . In: Proceedings of the 7th International Joint Conference on Automated Reasoning, pp. 480–494 (2014)Google Scholar
  9. 9.
    Chavira, M., Darwiche, A.: Compiling Bayesian networks with local structure. In: Proceedings of the Nineteenth International Joint Conference on Artificial Intelligence, pp. 1306–1312 (2005)Google Scholar
  10. 10.
    Cozman, F.G., Mauá, D.D.: Bayesian networks specified using propositional and relational constructs: combined, data, and domain complexity. In: Proceedings of the 29th AAAI Conference on Artificial Intelligence, pp. 3519–3525 (2015)Google Scholar
  11. 11.
    Dagum, P., Luby, M.: Approximating probabilistic inference in Bayesian belief networks is NP-hard. Artif. Intell. 60(1), 141–53 (1993)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    d’Amato, C., Fanizzi, N., Lukasiewicz, T.: Tractable reasoning with Bayesian description logics. In: Greco, S., Lukasiewicz, T. (eds.) SUM 2008. LNCS (LNAI), vol. 5291, pp. 146–159. Springer, Heidelberg (2008) CrossRefGoogle Scholar
  13. 13.
    D’Ambrosio, B.: Local expression languages for probabilistic dependence. Int. J. Approximate Reasoning 13(1), 61–1 (1995)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Darwiche, A.: Modeling and Reasoning with Bayesian Networks. Cambridge University Press, Cambridge (2009)zbMATHCrossRefGoogle Scholar
  15. 15.
    Domingos, P., Webb, W.: A tractable first-order probabilistic logic. In: Proceedings of the AAAI Conference on Artificial Intelligence (2012)Google Scholar
  16. 16.
    Friedman, N., Getoor, L., Koller, D., Pfeffer, A.: Learning probabilistic relational models. In: Proceedings of the International Joint Conference on Artificial Intelligence, pp. 1300–1309 (1999)Google Scholar
  17. 17.
    Getoor, L., Taskar, B.: Introduction to Statistical Relational Learning. MIT Press, Cambridge (2007)zbMATHGoogle Scholar
  18. 18.
    Halpern, J.Y.: Reasoning About Uncertainty. MIT Press, Cambridge (2003)zbMATHGoogle Scholar
  19. 19.
    Heckerman, D.: A tractable inference algorithm for diagnosing multiple diseases. In: Proceedings of the 5th Conference on Uncertainty in Artificial Intelligence, pp. 174–181 (1989)Google Scholar
  20. 20.
    Heinsohn, J.: Probabilistic description logics. In: Proceedings of the 10th International Conference on Uncertainty in Artificial Intelligence, pp. 311–318 (1994)Google Scholar
  21. 21.
    Jaeger, M.: Relational Bayesian networks. In: Procedings of the Conference on Uncertainty in Artificial Intelligence, pp. 266–273 (1997)Google Scholar
  22. 22.
    Jaeger, M.: Complex probabilistic modeling with recursive relational Bayesian networks. Ann. Math. Artif. Intell. 32, 179–220 (2001)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Jaeger, M., van Den Broeck, G.: Liftability of probabilistic inference: upper and lower bounds. In: Proceedings of the 2nd Statistical Relational AI Workshop (2012)Google Scholar
  24. 24.
    Koller, D., Friedman, N.: Probabilistic Graphical Models. MIT press, Cambridge (2009)zbMATHGoogle Scholar
  25. 25.
    Kwisthout, J.: Treewidth and the computational complexity of MAP approximations. In: van der Gaag, L.C., Feelders, A.J. (eds.) PGM 2014. LNCS, vol. 8754, pp. 271–285. Springer, Heidelberg (2014) Google Scholar
  26. 26.
    Kwisthout, J.H.P., Bodlaender, H.L., van der Gaag, L.C.: The necessity of bounded treewidth for efficient inference in Bayesian networks. In: Proceedings of the 19th European Conference on Artificial Intelligence, pp. 237–242 (2010)Google Scholar
  27. 27.
    Lin, C., Liu, J., Lu, P.: A simple FPTAS for counting edge covers. In: Proceedings of the 8th Annual ACM-SIAM Symposium on Discrete Algorightms, pp. 341–348 (2014)Google Scholar
  28. 28.
    Liu, J., Lu, P., Zhang, C.: FPTAS for counting weighted edge covers. In: Proceedings of the 22nd Annual European Symposium on Algorithms, pp. 654–665 (2014)Google Scholar
  29. 29.
    Lowd, D., Rooshenas, A.: Learning Markov networks with arithmetic circuits. In: Proceedings of the 16th International Conference on Artificial Intelligence and Statistics, pp. 406–414 (2013)Google Scholar
  30. 30.
    Mauá, D.D., Cozman, F.G.: A tractable class of model counting problems. Technical report, Decision Making Laboratory, University of São Paulo (2015)Google Scholar
  31. 31.
    Poon, H., Domingos, P.: Sum-product networks: a new deep architecture. In: Proceedings of the Twenty-Seventh Conference on Uncertainty in Artificial Intelligence, pp. 337–346 (2011)Google Scholar
  32. 32.
    De Raedt, L.: Logical and Relational Learning. Springer, New York (2008)zbMATHCrossRefGoogle Scholar
  33. 33.
    De Raedt, L., Kersting, K.: Probabilistic inductive logic programming. In: De Raedt, L., Frasconi, P., Kersting, K., Muggleton, S.H. (eds.) Probabilistic Inductive Logic Programming. LNCS (LNAI), vol. 4911, pp. 1–27. Springer, Heidelberg (2008) CrossRefGoogle Scholar
  34. 34.
    Ramachandran, R., Qi, G., Wang, K., Wang, J., Thornton, J.: Probabilistic reasoning in DL-Lite. In: Proceedings of the 12th Pacific Rim International Conference on Trends in Artificial Intelligence, pp. 480–491 (2012)Google Scholar
  35. 35.
    Rosenkrantz, D.J., Marathe, M.V., s. Ravi, S., Vullikanti, A.K.: Bayesian inference in treewidth-bounded graphical models without indegree constraints. In: Proceedings of the 30th Conference on Uncertainty in Artificial Intelligence, pp. 702–711 (2014)Google Scholar
  36. 36.
    Roth, D.: On the hardness of approximate reasoning. Artif. Intell. 82(1–2), 273–302 (1996)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Universidade de São PauloSão PauloBrazil

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