Local structure learning of chain graphs with the false discovery rate control

  • Jingyun Wang
  • Sanyang Liu
  • Mingmin Zhu


Chain graphs (CGs) containing both directed and undirected edges, offer an elegant generalisation of both Markov networks and Bayesian networks. In this paper, we propose an algorithm for local structure learning of CGs. It works by first learning adjacent nodes of each variable for skeleton identification and then orienting the edges of the complexes of the graph. To control the false discovery rate (FDR) of edges when learning a CG, FDR controlling procedure is embedded in the algorithm. Algorithms for skeleton identification and complexes recovery are presented. Experimental results demonstrate that the algorithm with the FDR controlling procedure can control the false discovery rate of the skeleton of the recovered graph under a user-specified level, and the proposed algorithm is also a viable alternative to learn the structure of chain graphs.


Chain graph Markov property False discovery rate Structure learning 



The research is supported by the National Natural Science Foundation of China (Grant No. 61373174) and (Grant No. 11401454).


  1. Andersson SA, Madigan D, Perlman MD (1996) An alternative Markov property for chain graphs. In: Proceedings of the twelfth international conference on uncertainty in artificial intelligence, pp 40–48Google Scholar
  2. Benjamini Y, Hochberg Y (1995) Controlling the false discovery rate: a practical and powerful approach to multiple testing. J R Stat Soc Ser B (Methodol) 57(1):289–300MathSciNetzbMATHGoogle Scholar
  3. Benjamini Y, Yekutieli D (2001) The control of the false discovery rate in multiple testing under dependency. Ann Stat 29(4):1165–1188MathSciNetCrossRefGoogle Scholar
  4. Bockhorst J, Craven M (2004) Markov networks for detecting overlapping elements in sequence data. In: Proceedings of the 17th international conference on neural information processing systems, pp 193–200Google Scholar
  5. Cai B, Liu Y, Fan Q, Zhang Y, Liu Z, Shilin Y, Ji R (2014) Multi-source information fusion based fault diagnosis of ground-source heat pump using Bayesian network. Appl Energy 114:1–9CrossRefGoogle Scholar
  6. Cai B, Liu Y, Fan Q, Zhang Y, Liu Z, Yu S, Ji R (2016) A multiphase dynamic Bayesian networks methodology for the determination of safety integrity levels. Reliab Eng Syst Saf 150:105–115CrossRefGoogle Scholar
  7. Cowell RG, Philip Dawid A, Lauritzen SL, Spiegelhalter DJ (2001) Probabilistic networks and expert systems. Publ Am Stat Assoc 43(1):108–109zbMATHGoogle Scholar
  8. Cox DR, Wermuth N (1993) Linear dependencies represented by chain graphs. Stat Sci 8(3):204–218MathSciNetCrossRefGoogle Scholar
  9. Cox DR, Wermuth N (1996) Multivariate dependencies: models, analysis and interpretation. Chapman and Hall, LondonzbMATHGoogle Scholar
  10. Flammini F, Marrone S, Mazzocca N, Nardone R, Vittorini V (2015) Using Bayesian networks to evaluate the trustworthiness of 2 out of 3 decision fusion mechanisms in multi-sensor applications. IFAC Papersonline 48(21):682–687CrossRefGoogle Scholar
  11. Frydenberg M (1990) The chain graph Markov property. Scand J Stat 17(4):333–353MathSciNetzbMATHGoogle Scholar
  12. Guo X, Zhang J, Cai Z, Du DZ, Pan Y (2015) DAM: a Bayesian method for detecting genome-wide associations on multiple diseases. Springer, BerlinGoogle Scholar
  13. Jayech K, Mahjoub MA (2011) Clustering and Bayesian network for image of faces classification. Int J Adv Comput Sci Appl 1(1):35–44Google Scholar
  14. Lauritzen SL, Wermuth N (1989) Graphical models for associations between variables, some of which are qualitative and some quantitative. Ann Stat 17(1):31–57MathSciNetCrossRefGoogle Scholar
  15. Listgarten J, Heckerman D (2007) Determining the number of non-spurious arcs in a learned DAG model: investigation of a Bayesian and a frequentist approach. In: Proceedings of the conference on uncertainty in artificial intelligence, pp 251–258Google Scholar
  16. Ma Z, Xie X, Geng Z (2008) Structural learning of chain graphs via decomposition. J Mach Learn Res 9(9):2847–2880MathSciNetzbMATHGoogle Scholar
  17. Margaritis D, Thrun S (1999) Bayesian network induction via local neighborhoods. Adv Neural Inf Process Syst 12:505–511Google Scholar
  18. Nielsen JD (2002) On local optima in learning Bayesian networks. In: Proceedings of the nineteenth conference on uncertainty in artificial intelligence, pp 435–442Google Scholar
  19. Peña JM (2009) Faithfulness in chain graphs: the discrete case. Int J Approx Reason 50(8):1306–1313MathSciNetCrossRefGoogle Scholar
  20. Peña JM (2011) Faithfulness in chain graphs: the Gaussian case. In: Proceedings of the 14th international conference on artificial intelligence and statistics, pp 588–599Google Scholar
  21. Peña JM, Nilsson R, Björkegren J, Tegnér J (2007) Towards scalable and data efficient learning of Markov boundaries. Int J Approx Reason 45(2):211–232CrossRefGoogle Scholar
  22. Peña JM, Sonntag D, Nielsen J (2014) An inclusion optimal algorithm for chain graph structure learning. In: Proceedings of the 17th international conference on artificial intelligence and statistics, pp 778–786Google Scholar
  23. Salama KM, Freitas AA (2013) ACO-based Bayesian network ensembles for the hierarchical classification of ageing-related proteins. In: Proceedings of the European conference on evolutionary computation, machine learning and data mining in bioinformatics, pp 80–91CrossRefGoogle Scholar
  24. Sonntag D, Peña JM (2015) Chain graph interpretations and their relations revisited. Int J Approx Reason 58:39–56MathSciNetCrossRefGoogle Scholar
  25. Sonntag D, Järvisalo M, Peña JM, Hyttinen A (2015a) Learning optimal chain graphs with answer set programming. In: Proceedings of the conference on uncertainty in artificial intelligence, pp 822–831Google Scholar
  26. Sonntag D, Peña JM, Gómez-Olmedo M (2015b) Approximate counting of graphical models via MCMC revisited. Int J Intell Syst 30(3):384–420CrossRefGoogle Scholar
  27. Studenỳ M (1997) A recovery algorithm for chain graphs. Int J Approx Reason 17(2–3):265–293MathSciNetCrossRefGoogle Scholar
  28. Triebel R, Kersting K, Burgard W (2006) Robust 3d scan point classification using associative Markov networks. In: IEEE international conference on robotics and automation, ICRA 2006, pp 2603–2608Google Scholar
  29. Tsamardinos I, Aliferis CF, Statnikov A (2003a) Time and sample efficient discovery of Markov blankets and direct causal relations. In: Proceedings of the international conference on knowledge discovery and data mining, pp 673–678Google Scholar
  30. Tsamardinos I, Aliferis CF, Statnikov AR (2003b) Algorithms for large scale Markov blanket discovery. In: Proceedings of the international flairs conference, pp 376–380Google Scholar
  31. Weber P, Medina-Oliva G, Simon C, Iung B (2012) Overview on Bayesian networks applications for dependability, risk analysis and maintenance areas. Eng Appl Artif Intell 25(4):671–682CrossRefGoogle Scholar
  32. Zhang L, Ji Q (2011) A Bayesian network model for automatic and interactive image segmentation. IEEE Trans Image Process 20(9):2582–2593MathSciNetCrossRefGoogle Scholar
  33. Zhang L, Zeng Z, Ji Q (2011) Probabilistic image modeling with an extended chain graph for human activity recognition and image segmentation. IEEE Trans Image Process 20(9):2401–2413MathSciNetCrossRefGoogle Scholar
  34. Zuo J, Wang M, Wan J, Genxiu W, Shuixiu W (2005) Modified information retrieval model based on Markov network. J Tsinghua Univ 345(3):307–314Google Scholar

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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsXidian UniversityXi’anChina

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