Skip to main content

Measures for the Evaluation and Comparison of Graphical Model Structures

  • Conference paper
  • First Online:
Computer Aided Systems Theory – EUROCAST 2017 (EUROCAST 2017)


Structure learning is the identification of the structure of graphical models based solely on observational data and is NP-hard. An important component of many structure learning algorithms are heuristics or bounds to reduce the size of the search space. We argue that variable relevance rankings that can be easily calculated for many standard regression models can be used to improve the efficiency of structure learning algorithms. In this contribution, we describe measures that can be used to evaluate the quality of variable relevance rankings, especially the well-known normalized discounted cumulative gain (NDCG). We evaluate and compare different regression methods using the proposed measures and a set of linear and non-linear benchmark problems.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others


  1. Campos, C.P., Ji, Q.: Efficient structure learning of Bayesian networks using constraints. J. Mach. Learn. Res. 12, 663–689 (2011)

    MathSciNet  MATH  Google Scholar 

  2. Chickering, D.M.: Learning Bayesian Networks is NP-Complete, pp. 121–130. Springer, Heidelberg (1996).

    Google Scholar 

  3. Elidan, G., Nachman, I., Friedman, N.: “Ideal Parent” structure learning for continuous variable Bayesian networks. J. Mach. Learn. Res. 8(8), 1799–1833 (2007)

    MathSciNet  MATH  Google Scholar 

  4. Friedman, J., Hastie, T., Tibshirani, R.: Sparse inverse covariance estimation with the graphical lasso. Biostatistics 9(3), 432–441 (2008)

    Article  MATH  Google Scholar 

  5. Friedman, N., Nachman, I.: Gaussian process networks. In: Proceedings of the Sixteenth Conference on Uncertainty in Artificial Intelligence (UAI), pp. 211–219. Morgan Kaufmann Publishers (2000)

    Google Scholar 

  6. Hand, D.J., Till, R.J.: A simple generalisation of the area under the ROC curve for multiple class classification problems. Mach. Learn. 45(2), 171–186 (2001).

    Article  MATH  Google Scholar 

  7. Hofmann, R., Tresp, V.: Discovering structure in continuous variables using Bayesian networks. In: Advances in Neural Information Processing Systems (NIPS), pp. 500–506 (1996)

    Google Scholar 

  8. Järvelin, K., Kekäläinen, J.: Cumulated gain-based evaluation of IR techniques. ACM Trans. Inf. Syst. (TOIS) 20(4), 422–446 (2002)

    Article  Google Scholar 

  9. Koivisto, M., Sood, K.: Exact Bayesian structure discovery in Bayesian networks. J. Mach. Learn. Res. 5(May), 549–573 (2004)

    MathSciNet  MATH  Google Scholar 

  10. Koller, D., Friedman, N.: Probabilistic Graphical Models: Principles and Techniques - Adaptive Computation and Machine Learning. MIT Press, Cambridge (2009)

    Google Scholar 

  11. Koski, T.J., Noble, J.: A review of Bayesian networks and structure learning. Math. Applicanda 40(1), 51–103 (2012)

    MathSciNet  MATH  Google Scholar 

  12. Kronberger, G.: Symbolic Regression for Knowledge Discovery - Bloat, Overfitting, and Variable Interaction Networks. Reihe C: Technik und Naturwissenschaften, Trauner Verlag (2011)

    Google Scholar 

  13. Kronberger, G., Fink, S., Kommenda, M., Affenzeller, M.: Macro-economic time series modeling and interaction networks. In: Di Chio, C., Brabazon, A., Di Caro, G.A., Drechsler, R., Farooq, M., Grahl, J., Greenfield, G., Prins, C., Romero, J., Squillero, G., Tarantino, E., Tettamanzi, A.G.B., Urquhart, N., Uyar, A.Ş. (eds.) EvoApplications 2011. LNCS, vol. 6625, pp. 101–110. Springer, Heidelberg (2011).

    Chapter  Google Scholar 

  14. Meinshausen, N., Bühlmann, P.: High-dimensional graphs and variable selection with the lasso. Annal. Stat. 34(3), 1436–1462 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Rao, R., Lakshminarayanan, S.: Variable interaction network based variable selection for multivariate calibration. Anal. Chim. Acta 599(1), 24–35 (2007)

    Article  Google Scholar 

  16. Singh, A.P., Moore, A.W.: Finding optimal Bayesian networks by dynamic programming. Technical report CMU-CALD-05-1062, School of Computer Science, Carnegie Mellon University, June 2005

    Google Scholar 

  17. Spirtes, P., Glymour, C., Scheines, R.: Causation, Prediction, and Search. Springer, New York (1993).

    Book  MATH  Google Scholar 

  18. Tsamardinos, I., Brown, L.E., Aliferis, C.F.: The max-min hill-climbing Bayesian network structure learning algorithm. Mach. Learn. 65(1), 31–78 (2006)

    Article  Google Scholar 

  19. Winker, S., Affenzeller, M., Kronberger, G., Kommenda, M., Wagner, S., Jacak, W., Stekel, H.: Variable interaction networks in medical data. In: Proceedings of the 24th European Modeling and Simulation Symposium EMSS 2012, pp. 265–270. Dime Universitá di Genova (2012)

    Google Scholar 

Download references


The authors gratefully acknowledge financial support by the Austrian Research Promotion Agency (FFG) and the Government of Upper Austria within the COMET Project #843532 Heuristic Optimization in Production and Logistics (HOPL).

Author information

Authors and Affiliations


Corresponding author

Correspondence to Gabriel Kronberger .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Kronberger, G., Burlacu, B., Kommenda, M., Winkler, S., Affenzeller, M. (2018). Measures for the Evaluation and Comparison of Graphical Model Structures. In: Moreno-Díaz, R., Pichler, F., Quesada-Arencibia, A. (eds) Computer Aided Systems Theory – EUROCAST 2017. EUROCAST 2017. Lecture Notes in Computer Science(), vol 10671. Springer, Cham.

Download citation

  • DOI:

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-74717-0

  • Online ISBN: 978-3-319-74718-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics