Bayesian Networks for Max-Linear Models

  • Claudia KlüppelbergEmail author
  • Steffen Lauritzen


We study Bayesian networks based on max-linear structural equations as introduced in Gissibl and Klüppelberg (2018) and provide a summary of their independence properties. In particular, we emphasize that distributions for such networks are generally not faithful to the independence model determined by their associated directed acyclic graph. In addition, we consider some of the basic issues of estimation and discuss generalized maximum likelihood estimation of the coefficients, using the concept of a generalized likelihood ratio for non-dominated families as introduced by Kiefer and Wolfowitz (1956). Finally, we argue that the structure of a minimal network asymptotically can be identified completely from observational data.



The authors have benefited from discussions with Nadine Gissibl and financial support from the Alexander von Humboldt Stiftung.


  1. Asadi, P., Davison, A. C. & Engelke, S. (2015), ‘Extremes on river networks’, Ann. Appl. Stat.9(4), 2023–2050.
  2. Beirlant, J., Goegebeur, Y., Segers, J., Teugels, J., De Waal, D. & Ferro, C. (2006), Statistics of Extremes: Theory and Applications, Wiley, Chichester.zbMATHGoogle Scholar
  3. Bollen, K. A. (1989), Structural Equations with Latent Variables, Wiley, New York.CrossRefGoogle Scholar
  4. Buhl, S., Davis, R. A., Klüppelberg, C. & Steinkohl, C. (2016), ‘Semiparametric estimation for isotropic max-stable space-time processes’, arXiv:1609.04967.Google Scholar
  5. Butkovič, P. (2010) , Max-linear Systems: Theory and Algorithms, Springer, London.CrossRefGoogle Scholar
  6. Davis, R. A., Klüppelberg, C. & Steinkohl, C. (2013), ‘Statistical inference for max-stable processes in space and time’, Journal of the Royal Statistical Society: Series B (Statistical Methodology)75(5), 791–819.Google Scholar
  7. Davis, R. A. & Resnick, S. I. (1989) , ‘Basic properties and prediction of max-ARMA processes’, Advances in Applied Probability21(4), 781–803.Google Scholar
  8. Davison, A. C., Padoan, S. A. & Ribatet, M. (2012), ‘Statistical modeling of spatial extremes’, Statist. Sci.27(2), 161–186.
  9. Dawid, A. P. (1980), ‘Conditional independence for statistical operations’, The Annals of Statistics8(3), 598–617.MathSciNetCrossRefGoogle Scholar
  10. de Haan, L. & Ferreira, A. (2006), Extreme Value Theory: An Introduction, Springer, New York.CrossRefGoogle Scholar
  11. Einmahl, J. H. J., Kiriliouk, A. & Segers, J. (2018), ‘A continuous updating weighted least squares estimator of tail dependence in high dimensions’, Extremes21(2), 205–233.
  12. Embrechts, P., Klüppelberg, C. & Mikosch, T. (1997), Modelling Extremal Events: for Insurance and Finance, Springer-Verlag Berlin Heidelberg.CrossRefGoogle Scholar
  13. Finkenstädt, B. & Rootzén, H. (2004), Extreme Values in Finance, Telecommunications, and the Environment, Chapman & Hall/CRC, Boca Raton.zbMATHGoogle Scholar
  14. Frydenberg, M. (1990), ‘The chain graph Markov property’, Scandinavian Journal of Statistics17(4), 333–353.MathSciNetzbMATHGoogle Scholar
  15. Gill, R. D., Wellner, J. A. & Præstgaard, J. (1989) , ‘Non- and semi-parametric maximum likelihood estimators and the von Mises method (part 1)’, Scandinavian Journal of Statistics16(2), 97–128.MathSciNetGoogle Scholar
  16. Gissibl, N. & Klüppelberg, C. (2018), ‘Max-linear models on directed acyclic graphs’, Bernoulli24(4A), 2693–2720.
  17. Gissibl, N., Klüppelberg, C. & Lauritzen, S. (2019), ‘Identifiability and estimation of recursive max-linear models’, arXiv:1901.03556Google Scholar
  18. Gissibl, N., Klüppelberg, C. & Mager, J. (2017), Big data: Progress in automating extreme risk analysis, in W. Pietsch, J. Wernecke & M. Ott, eds, ‘Berechenbarkeit der Welt? Philosophie und Wissenschaft im Zeitalter von Big Data’, Springer Fachmedien Wiesbaden, Wiesbaden, pp. 171–189.CrossRefGoogle Scholar
  19. Hoef, J. M. V., Peterson, E. E. & Theobald, D. (2006), ‘Spatial statistical models that use flow and stream distance’, Environmental and Ecological Statistics13(4), 449–464.MathSciNetCrossRefGoogle Scholar
  20. Huser, R. & Davison, A. C. (2014), ‘Space-time modelling of extreme events’, Journal of the Royal Statistical Society: Series B (Statistical Methodology)76(2), 439–461.MathSciNetCrossRefGoogle Scholar
  21. Johansen, S. (1978) , ‘The product limit estimator as maximum likelihood estimator’, Scandinavian Journal of Statistics5(4), 195–199.MathSciNetzbMATHGoogle Scholar
  22. Kiefer, J. & Wolfowitz, J. (1956), ‘Consistency of the maximum likelihood estimator in the presence of infinitely many incidental parameters’, The Annals of Mathematical Statistics27(4), 887–906.MathSciNetCrossRefGoogle Scholar
  23. Lauritzen, S. & Sadeghi, K. (2018), ‘Unifying Markov properties for graphical models’, Ann. Statist.46(5), 2251–2278.
  24. Lauritzen, S. (1996), Graphical Models, Oxford University Press, Oxford.zbMATHGoogle Scholar
  25. Lauritzen, S. L., Dawid, A. P., Larsen, B. N. & Leimer, H.-G. (1990), ‘Independence properties of directed Markov fields’, Networks20(5), 491–505.MathSciNetCrossRefGoogle Scholar
  26. Meek, C. (1995), Strong completeness and faithfulness in Bayesian networks, in P. Besnard & S. Hanks, eds, ‘Proceedings of the 11th Conference on Uncertainty in Artificial Intelligence’, Morgan Kaufmann Publishers, San Francisco, pp. 411–418.Google Scholar
  27. Peters, J. & Bühlmann, P. (2014), ‘Identifiability of Gaussian structural equation models with equal error variances’, Biometrika101(1), 219–228.Google Scholar
  28. Resnick, S. I. (1987), Extreme Values, Regular Variation, and Point Processes, Springer, New York.CrossRefGoogle Scholar
  29. Resnick, S. I. (2007), Heavy-Tail Phenomena: Probabilistic and Statistical Modeling, Springer, New York.zbMATHGoogle Scholar
  30. Richardson, T. (2003), ‘Markov properties for acyclic directed mixed graphs’, Scandinavian Journal of Statistics30(1), 145–157.MathSciNetCrossRefGoogle Scholar
  31. Scholz, F. W. (1980) , ‘Towards a unified definition of maximum likelihood’, The Canadian Journal of Statistics8(2), 193–203.MathSciNetCrossRefGoogle Scholar
  32. Shimizu, S., Hoyer, P. O., Hyvärinen, A. & Kerminen, A. (2006) , ‘A linear non-Gaussian acyclic model for causal discovery’, J. Mach. Learn. Res.7, 2003–2030.MathSciNetzbMATHGoogle Scholar
  33. Spirtes, P., Glymour, C. & Scheines, R. (2000) , Causation, Prediction, and Search, 2nd edn, MIT press.Google Scholar
  34. Spirtes, P. & Zhang, J. (2014) , ‘A uniformly consistent estimator of causal effects under the \(k\)-triangle-faithfulness assumption’, Statist. Sci.29(4), 662–678.
  35. Verma, T. & Pearl, J. (1990), Equivalence and synthesis of causal models, in L. N. K. P. Bonissone, M. Henrion & J. F. Lemmer, eds, ‘Proceedings of the 6th Conference on Uncertainty in Artificial Intelligence’, Amsterdam, pp. 255–270.Google Scholar
  36. Wang, Y. & Stoev, S. A. (2011), ‘Conditional sampling for spectrally discrete max-stable random fields’, Adv. in Appl. Probab.43(2), 461–483.MathSciNetCrossRefGoogle Scholar
  37. West, D. (2001), Introduction to Graph Theory, Prentice Hall.Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Center for Mathematical SciencesTechnical University of MunichMunichGermany
  2. 2.Department of Mathematical SciencesUniversity of CopenhagenCopenhagenDenmark

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