Abstract
Dempster-Shafer Theory (DST) generalizes Bayesian probability theory, offering useful additional information, but suffers from a high computational burden. A lot of work has been done to reduce the complexity of computations used in information fusion with Dempster’s rule. The main approaches exploit either the structure of Boolean lattices or the information contained in belief sources. Each has its merits depending on the situation. In this paper, we propose sequences of graphs for the computation of the zeta and Möbius transformations that optimally exploit both the structure of distributive lattices and the information contained in belief sources. We call them the Efficient Möbius Transformations (EMT). We show that the complexity of the EMT is always inferior to the complexity of algorithms that consider the whole lattice, such as the Fast Möbius Transform (FMT) for all DST transformations. We then explain how to use them to fuse two belief sources. More generally, our EMTs apply to any function in any finite distributive lattice, focusing on a meet-closed or join-closed subset.
This work was carried out and co-funded in the framework of the Labex MS2T and the Hauts-de-France region of France. It was supported by the French Government, through the program “Investments for the future” managed by the National Agency for Research (Reference ANR-11-IDEX-0004-02).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
The following definitions hold for lower semifinite partially ordered sets as well, i.e. partially ordered sets such that the number of elements of P lower in the sense of \(\le \) than another element of P is finite. But for the sake of simplicity, we will only talk of finite partially ordered sets.
- 2.
This unit cost can be obtained when \(P=2^\varOmega \) using a dynamic binary tree as data structure for the representation of M. With it, finding the proxy element only takes the reading of a binary string, considered as one operation. Further details will soon be available in an extended version of this work [5].
References
Barnett, J.A.: Computational methods for a mathematical theory of evidence. In: Proceedings of IJCAI, vol. 81, pp. 868–875 (1981)
Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M.: Trimmed moebius inversion and graphs of bounded degree. Theory Comput. Syst. 47(3), 637–654 (2010)
Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M., Nederlof, J., Parviainen, P.: Fast zeta transforms for lattices with few irreducibles. ACM TALG 12(1), 4 (2016)
Chaveroche, M., Davoine, F., Cherfaoui, V.: Calcul exact de faible complexité des décompositions conjonctive et disjonctive pour la fusion d’information. In: Proceedings of GRETSI (2019)
Chaveroche, M., Davoine, F., Cherfaoui, V.: Efficient algorithms for Möbius transformations and their applications to Dempster-Shafer Theory. Manuscript available on request (2019)
Dempster, A.: A generalization of Bayesian inference. J. Roy. Stat. Soc. Ser. B (Methodol.) 30, 205–232 (1968)
Gordon, J., Shortliffe, E.H.: A method for managing evidential reasoning in a hierarchical hypothesis space. Artif. Intell. 26(3), 323–357 (1985)
Kaski, P., Kohonen, J., Westerbäck, T.: Fast Möbius inversion in semimodular lattices and U-labelable posets. arXiv preprint arXiv:1603.03889 (2016)
Kennes, R.: Computational aspects of the Mobius transformation of graphs. IEEE Trans. Syst. Man Cybern. 22(2), 201–223 (1992)
Sarabi-Jamab, A., Araabi, B.N.: Information-based evaluation of approximation methods in Dempster-Shafer Theory. IJUFKS 24(04), 503–535 (2016)
Shafer, G.: A Mathematical Theory of Evidence. Princeton University Press, Princeton (1976)
Shafer, G., Logan, R.: Implementing Dempster’s rule for hierarchical evidence. Artif. Intell. 33(3), 271–298 (1987)
Shenoy, P.P., Shafer, G.: Propagating belief functions with local computations. IEEE Expert 1(3), 43–52 (1986)
Wilson, N.: Algorithms for Dempster-Shafer Theory. In: Kohlas, J., Moral, S. (eds.) Handbook of Defeasible Reasoning and Uncertainty Management Systems: Algorithms for Uncertainty and Defeasible Reasoning, vol. 5, pp. 421–475. Springer, Netherlands (2000). https://doi.org/10.1007/978-94-017-1737-3_10
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Chaveroche, M., Davoine, F., Cherfaoui, V. (2019). Efficient Möbius Transformations and Their Applications to D-S Theory. In: Ben Amor, N., Quost, B., Theobald, M. (eds) Scalable Uncertainty Management. SUM 2019. Lecture Notes in Computer Science(), vol 11940. Springer, Cham. https://doi.org/10.1007/978-3-030-35514-2_29
Download citation
DOI: https://doi.org/10.1007/978-3-030-35514-2_29
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-35513-5
Online ISBN: 978-3-030-35514-2
eBook Packages: Computer ScienceComputer Science (R0)