Skip to main content

A Priori Approximation of Symmetries in Dynamic Probabilistic Relational Models

Part of the Lecture Notes in Computer Science book series (LNAI,volume 12873)


Lifted inference approaches reduce computational work as inference is performed using representatives for sets of indistinguishable random variables, which allows for tractable inference w.r.t. domain sizes in dynamic probabilistic relational models. Unfortunately, maintaining a lifted representation is challenging in practically relevant application domains, as evidence often breaks symmetries making lifted techniques fall back on their ground counterparts. In existing approaches asymmetric evidence is counteracted by merging similar but distinguishable objects when moving forward in time. While undoing splits a posteriori is reasonable, we propose learning approximate model symmetries a priori to prevent unnecessary splits due to inaccuracy or one-time events. In particular, we propose a multivariate ordinal pattern symbolization approach followed by spectral clustering to determine sets of domain entities behaving approximately the same over time. By using object clusters, we avoid unnecessary splits by keeping entities together that tend to behave the same over time. Understanding symmetrical and asymmetrical entity behavior a priori allows for increasing accuracy in inference by means of inferred evidence for unobserved entities to better represent reality. Empirical results show that our approach reduces unnecessary splits, i.e., improves runtimes, while keeping accuracy in inference high.


  • Relational models
  • Lifting
  • Ordinal pattern
  • Symmetry

This is a preview of subscription content, access via your institution.

Buying options

USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-030-87626-5_23
  • Chapter length: 15 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
USD   59.99
Price excludes VAT (USA)
  • ISBN: 978-3-030-87626-5
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   79.99
Price excludes VAT (USA)
Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.


  1. 1.

    pronounced deeper models.

  2. 2.

  3. 3.


  1. Bandt, C., Pompe, B.: Permutation entropy: a natural complexity measure for time series. Phys. Rev. Lett. 88(17) (2002).

  2. Bellman, R.: Adaptive Control Processes: A Guided Tour. Princeton legacy library, Princeton University Press (2015).

  3. Bertozzi, A.L., Merkurjev, E.: Chapter 12 - Graph-based optimization approaches for machine learning, uncertainty quantification and networks. In: Kimmel, R., Tai, X.C. (eds.) Processing, Analyzing and Learning of Images, Shapes, and Forms: Part 2, Handbook of Numerical Analysis, vol. 20, pp. 503–531. Elsevier (2019).,

  4. Van den Broeck, G., Niepert, M.: Lifted probabilistic inference for asymmetric graphical models. In: Proceedings of the 29th Conference on Artificial Intelligence (AAAI) (2015)

    Google Scholar 

  5. Chiu, B., Keogh, E., Lonardi, S.: Probabilistic discovery of time series motifs. In: Proceedings of the Ninth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. KDD 2003, New York, NY, USA, pp. 493–498 (2003)

    Google Scholar 

  6. Finke, N., Gehrke, M., Braun, T., Potten, T., Möller, R.: Investigating matureness of probabilistic graphical models for dry-bulk shipping. In: Jaeger, M., Nielsen, T.D. (eds.) Proceedings of the 10th International Conference on Probabilistic Graphical Models. Proceedings of Machine Learning Research, vol. 138, pp. 197–208. PMLR, 23–25 September 2020

    Google Scholar 

  7. Gehrke, M., Braun, T., Möller, R.: Lifted dynamic junction tree algorithm. In: Chapman, P., Endres, D., Pernelle, N. (eds.) ICCS 2018. LNCS (LNAI), vol. 10872, pp. 55–69. Springer, Cham (2018).

    CrossRef  Google Scholar 

  8. Gehrke, M., Möller, R., Braun, T.: Taming reasoning in temporal probabilistic relational models. In: Proceedings of the 24th European Conference on Artificial Intelligence (ECAI 2020) (2020).

  9. Keller, K., Maksymenko, S., Stolz, I.: Entropy determination based on the ordinal structure of a dynamical system. Discrete Continuous Dyn. Syst. Ser. B 20(10), 3507–3524 (2015).

    MathSciNet  CrossRef  MATH  Google Scholar 

  10. Keller, K., Mangold, T., Stolz, I., Werner, J.: Permutation entropy: new ideas and challenges. Entropy 19(3), 134 (2017).

    CrossRef  Google Scholar 

  11. Kimmig, A., Mihalkova, L., Getoor, L.: Lifted graphical models: a survey. Mach. Learn. 99(1), 1–45 (2014).

    MathSciNet  CrossRef  MATH  Google Scholar 

  12. Mohr, M., Wilhelm, F., Hartwig, M., Möller, R., Keller, K.: New approaches in ordinal pattern representations for multivariate time series. In: Proceedings of the 33rd International Florida Artificial Intelligence Research Society Conference (FLAIRS-33), pp. 124–129. AAAI Press (2020)

    Google Scholar 

  13. Myers, A., Khasawneh, F.A.: On the automatic parameter selection for permutation entropy. Chaos Interdiscip. Jo. Nonlinear Sci. 30(3), 033130 (2020).,, publisher: American Institute of Physics

  14. Niepert, M., Van den Broeck, G.: Tractability through exchangeability: a new perspective on efficient probabilistic inference. In: AAAI-14 Proceedings of the 28th AAAI Conference on Artificial Intelligence, pp. 2467–2475. AAAI Press (2014)

    Google Scholar 

  15. Piek, A.B., Stolz, I., Keller, K.: Algorithmics, possibilities and limits of ordinal pattern based entropies. Entropy 21(6), 547 (2019).

    MathSciNet  CrossRef  Google Scholar 

  16. Poole, D.: First-order probabilistic inference. In: Proceedings of the 18th International Joint Conference on Artificial Intelligence, pp. 985–991. IJCAI Organization (2003)

    Google Scholar 

  17. Riedl, M., Müller, A., Wessel, N.: Practical considerations of permutation entropy: a tutorial review. Eur. Phys. J. Spec. Top. 222 (2013).

  18. Singla, P., Nath, A., Domingos, P.: Approximate lifting techniques for belief propagation. In: Proceedings of the Twenty-Eighth AAAI Conference on Artificial Intelligence, pp. 2497–2504. AAAI 2014. AAAI Press (2014)

    Google Scholar 

  19. Stolz, I., Keller, K.: A general symbolic approach to Kolmogorov-Sinai entropy. Entropy 19(12), 675 (2017).

    MathSciNet  CrossRef  Google Scholar 

  20. Venugopal, D., Gogate, V.: Evidence-based clustering for scalable inference in Markov logic. In: Calders, T., Esposito, F., Hüllermeier, E., Meo, R. (eds.) ECML PKDD 2014. LNCS (LNAI), vol. 8726, pp. 258–273. Springer, Heidelberg (2014).

    CrossRef  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Nils Finke .

Editor information

Editors and Affiliations



figure a
figure b

Rights and permissions

Reprints and Permissions

Copyright information

© 2021 Springer Nature Switzerland AG

About this paper

Verify currency and authenticity via CrossMark

Cite this paper

Finke, N., Mohr, M. (2021). A Priori Approximation of Symmetries in Dynamic Probabilistic Relational Models. In: Edelkamp, S., Möller, R., Rueckert, E. (eds) KI 2021: Advances in Artificial Intelligence. KI 2021. Lecture Notes in Computer Science(), vol 12873. Springer, Cham.

Download citation

  • DOI:

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-87625-8

  • Online ISBN: 978-3-030-87626-5

  • eBook Packages: Computer ScienceComputer Science (R0)