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Constraint-based learning for non-parametric continuous bayesian networks

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Abstract

Modeling high-dimensional multivariate distributions is a computationally challenging task. In the discrete case, Bayesian networks have been successfully used to reduce the complexity and to simplify the problem. However, they lack of a general model for continuous variables. In order to overcome this problem, Elidan (2010) proposed the model of copula Bayesian networks that parametrizes Bayesian networks using copula functions. We propose a new learning algorithm for this model based on a PC algorithm and a conditional independence test proposed by Bouezmarni et al. (2009). This test being non-parametric, no model assumptions are made allowing it to be as general as possible. This algorithm is compared on generated data with the parametric method proposed by Elidan (2010) and proves to have better results.

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Code Availability

The source code for CBIC and CPC methods is available on the GitHub repository openturns/otagrum.

References

  1. Baudin, M., Dutfoy, A., Iooss, B., Popelin, A.L.: Openturns: An industrial software for uncertainty quantification in simulation (2015)

  2. Bedford, T., Cooke, R.M., et al.: Vines–a new graphical model for dependent random variables. Ann. Stat. 30(4), 1031–1068 (2002)

    Article  MathSciNet  Google Scholar 

  3. Beinlich, I.A., Suermondt, H.J., Chavez, R.M., Cooper, G.F.: The alarm monitoring system: A case study with two probabilistic inference techniques for belief networks. In: AIME 89, pp 247–256. Springer (1989)

  4. Bouezmarni, T., Rombouts, J., Taamouti, A.: A nonparametric copula based test for conditional independence with applications to granger causality. Economics Working Papers. Universidad Carlos III, Departamento de Economía (2009)

  5. Bouezmarni, T., Rombouts, J.V., Taamouti, A.: Asymptotic properties of the bernstein density copula estimator for α-mixing data. J. Multivar. Anal. 101(1), 1–10 (2010). https://doi.org/10.1016/j.jmva.2009.02.014

    Article  MathSciNet  Google Scholar 

  6. Colombo, D., Maathuis, M.H.: Order-independent constraint-based causal structure learning. J. Machine Learn. Res. 15(1), 3741–3782 (2014)

    MathSciNet  MATH  Google Scholar 

  7. Czado, C.: Pair-copula constructions of multivariate copulas. In: Copula Theory and Its Applications, pp 93–109 (2010)

  8. Deheuvels, P.: La Fonction de dépendance Empirique et Ses Propriétés. Un test Non Paramétrique D’Indépendance. Bulletins de l’Académie Royale de Belgique 65(1), 274–292 (1979)

    MATH  Google Scholar 

  9. Elidan, G.: Copula bayesian networks. In: Advances in Neural Information Processing Systems, pp 559–567 (2010)

  10. Genest, C., Favre, A.C.: Everything you always wanted to know about copula modeling but were afraid to ask. J. Hydrol. Eng. 12(4), 347–368 (2007)

    Article  Google Scholar 

  11. Glover, F., Laguna, M.: Tabu Search, pp 2093–2229. Springer, Boston (1998). https://doi.org/10.1007/978-1-4613-0303-9_33

    MATH  Google Scholar 

  12. Gonzales, C., Torti, L., Wuillemin, P.H.: aGrUM: A graphical universal model framework. In: International Conference on Industrial Engineering, Other Applications of Applied Intelligent Systems, Proceedings of the 30th International Conference on Industrial Engineering, Other Applications of Applied Intelligent Systems. Arras, France. https://hal.archives-ouvertes.fr/hal-01509651 (2017)

  13. Huang, J.C.: Cumulative distribution networks: Inference, estimation and applications of graphical models for cumulative distribution functions. Citeseer (2009)

  14. Ide, J.S., Cozman, F.G.: Random generation of bayesian networks. In: Brazilian Symposium on Artificial Intelligence, pp 366–376. Springer (2002)

  15. Joe, H.: Multivariate models and multivariate dependence concepts. CRC Press (1997)

  16. Karra, K., Mili, L.: Hybrid copula bayesian networks. In: Conference on Probabilistic Graphical Models, pp 240–251 (2016)

  17. Koller, D., Friedman, N.: Probabilistic graphical models: principles and techniques. MIT Press (2009)

  18. Lasserre, M., Lebrun, R., Wuillemin, P.H.: Learning continuous high-dimensional models using mutual information and copula bayesian networks (2021)

  19. Lauritzen, S.L., Wermuth, N.: Graphical models for associations between variables, some of which are qualitative and some quantitative. Annals Stat. 31–57 (1989)

  20. Lindskog, F., McNeil, A., Schmock, U.: Kendall’s tau for elliptical distributions. In: Credit Risk, pp 149–156. Springer (2003)

  21. Nelsen, R.B.: An introduction to copulas. Springer Science & Business Media, New York (2007)

    MATH  Google Scholar 

  22. Rousseeuw, P.J., Molenberghs, G.: Transformation of non positive semidefinite correlation matrices. Commun. Stat. Theor. Methods 22(4), 965–984 (1993)

    Article  Google Scholar 

  23. Sancetta, A., Satchell, S.: The bernstein copula and its applications to modeling and approximations of multivariate distributions. Econ. Theor. 20(03), 535–562 (2004)

    Article  MathSciNet  Google Scholar 

  24. Schwarz, G.: Estimating the dimension of a model. Annals Stat. 6 (2), 461–464 (1978)

    Article  MathSciNet  Google Scholar 

  25. Sklar, A.: Fonctions de répartition à n dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris 8, 229–231 (1959)

    MathSciNet  MATH  Google Scholar 

  26. Spirtes, P., Glymour, C.N., Scheines, R., Heckerman, D., Meek, C., Cooper, G., Richardson, T.: Causation, prediction, and search. MIT Press (2000)

  27. Su, L., White, H.: A nonparametric hellinger metric test for conditional independence. Econ. Theor. 24(4), 829–864 (2008)

    Article  MathSciNet  Google Scholar 

  28. Wan, J., Zabaras, N.: A probabilistic graphical model based stochastic input model construction. J. Comput. Physics 272, 664–685 (2014)

    Article  MathSciNet  Google Scholar 

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Funding

This work was partially supported by Airbus Research through the AtRandom project (CRT/VPE/XRD).

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Author notes

  1. Marvin Lasserre & Pierre-Henri Wuillemin

    Present address: Sorbonne Université- LIP6, 4 place Jussieu, 75005, Paris, France

Authors and Affiliations

  1. Airbus AI Research, 22 rue du Gouverneur Général Eboué, 92130, Issy les Moulineaux, France

    Régis Lebrun

Authors
  1. Marvin Lasserre
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  2. Régis Lebrun
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  3. Pierre-Henri Wuillemin
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Correspondence to Marvin Lasserre.

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The authors declare that they have no conflict of interest.

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The source code to generate data and figure is provided on the GitHub repository MLasserre/otagrum-experiments.

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Lasserre, M., Lebrun, R. & Wuillemin, PH. Constraint-based learning for non-parametric continuous bayesian networks. Ann Math Artif Intell 89, 1035–1052 (2021). https://doi.org/10.1007/s10472-021-09754-2

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