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Measuring Quality of Belief Function Approximations

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Integrated Uncertainty in Knowledge Modelling and Decision Making (IUKM 2022)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 13199))

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Abstract

Because of the high computational complexity of the respective procedures, the application of belief-function theory to problems of practice is possible only when the considered belief functions are approximated in an efficient way. Not all measures of similarity/dissimilarity are felicitous to measure the quality of such approximations. The paper presents results from a pilot study that tries to detect the divergences suitable for this purpose.

Financially supported by the Czech National Science Foundation under grant no. 19-06569S.

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Notes

  1. 1.

    We take \(0 \log _2(0) = 0\).

  2. 2.

    To show asymmetry of the Kullback-Leibler divergence consider \(\mu = (\frac{1}{3}, \frac{1}{3},\frac{1}{3})\), and \(\kappa = (\frac{1}{2}, \frac{1}{2},0)\).

  3. 3.

    \(2^{\mathbb {X}_N}\) denote the set of all subsets of \({\mathbb {X}_N}\).

  4. 4.

    \(K_1, K_2, \ldots , K_n\) meets the running intersection property if

    $$\forall {i} = 2, 3, \ldots , n\ \ \exists \ j\ (1\le j < i)\ \ K_i \cap (K_1 \cup \ldots \cup K_{i-1}) \subseteq K_j. $$

    .

  5. 5.

    We generated basic assignments of three types: 300 of them were nested, 300 were quasi-bayesian, and the remaining 300 basic assignments had 29 fully randomly selected focal elements and the thirties one was \(\mathbb {X}_N\).

  6. 6.

    Precisely speaking, we know that all RIP models are perfect, but, theoretically, it may happen that also non-RIP model is perfect. However, this happens very rarely, and when assessing the results, we took that all non-RIP models were non-perfect.

References

  1. Cobb, B.R., Shenoy, P.P.: On the plausibility transformation method for translating belief function models to probability models. Int. J. Approximate Reasoning 41(3), 314–340 (2006)

    Article  MathSciNet  Google Scholar 

  2. Cuzzolin, F.: On the relative belief transform. Int. J. Approximate Reasoning 53(5), 786–804 (2012)

    Article  MathSciNet  Google Scholar 

  3. Daniel, M.: On transformations of belief functions to probabilities. Int. J. Intell. Syst. 21(3), 261–282 (2006)

    Article  Google Scholar 

  4. Jiroušek, R.: Foundations of compositional model theory. Int. J. Gen. Syst. 40(6), 623–678 (2011)

    Article  MathSciNet  Google Scholar 

  5. Jiroušek, R.: A short note on decomposition and composition of knowledge. Int. J. Approximate Reasoning 120, 24–32 (2020)

    Article  MathSciNet  Google Scholar 

  6. Jiroušek, R., Kratochvíl, V.: Approximations of belief functions using compositional models. In: Vejnarová, J., Wilson, N. (eds.) ECSQARU 2021. LNCS (LNAI), vol. 12897, pp. 354–366. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-86772-0_26

    Chapter  Google Scholar 

  7. Jiroušek, R., Kratochvíl, V., Shenoy, P.P.: Entropy-based learning of compositional models from data. In: Denœux, T., Lefèvre, E., Liu, Z., Pichon, F. (eds.) BELIEF 2021. LNCS (LNAI), vol. 12915, pp. 117–126. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-88601-1_12

    Chapter  Google Scholar 

  8. Jiroušek, R., Shenoy, P.P.: Compositional models in valuation-based systems. In: Denoeux, T., Masson, M.H. (eds.) Belief Functions: Theory and Applications. Advances in Intelligent and Soft Computing, pp. 221–228. Springer, Berlin, Heidelberg (2012). https://doi.org/10.1007/978-3-642-29461-7_26

    Chapter  Google Scholar 

  9. Jiroušek, R., Shenoy, P.P.: A new definition of entropy of belief functions in the Dempster-Shafer theory. Int. J. Approximate Reasoning 92(1), 49–65 (2018)

    Article  MathSciNet  Google Scholar 

  10. Jiroušek, R., Vejnarová, J., Daniel, M.: Compositional models for belief functions. In: de Cooman, G., Vejnarová, J., Zaffalon, M. (eds.) Proceedings of the Fifth International Symposium on Imprecise Probability: Theories and Applications (ISIPTA 2007), pp. 243–252 (2007)

    Google Scholar 

  11. Jousselme, A.L., Grenier, D., Bossé, É.: A new distance between two bodies of evidence. Inform. Fusion 2(2), 91–101 (2001)

    Article  Google Scholar 

  12. Jousselme, A.L., Maupin, P.: Distances in evidence theory: comprehensive survey and generalizations. Int. J. Approximate Reasoning 53(2), 118–145 (2012)

    Article  MathSciNet  Google Scholar 

  13. Kullback, S., Leibler, R.A.: On information and sufficiency. Ann. Math. Stat. 22, 76–86 (1951)

    Article  MathSciNet  Google Scholar 

  14. Österreicher, F., Vajda, I.: A new class of metric divergences on probability spaces and its applicability in statistics. Ann. Inst. Stat. Math. 55(3), 639–653 (2003). https://doi.org/10.1007/BF02517812

    Article  MathSciNet  MATH  Google Scholar 

  15. Perez, A.: \(\varepsilon \)-admissible simplifications of the dependence structure of a set of random variables. Kybernetika 13(6), 439–449 (1977)

    MathSciNet  MATH  Google Scholar 

  16. Shafer, G.: A Mathematical Theory of Evidence. Princeton University Press, Princeton (1976)

    Book  Google Scholar 

  17. Shannon, C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27(379–423), 623–656 (1948)

    Article  MathSciNet  Google Scholar 

  18. Smets, P.: Constructing the Pignistic probability function in a context of uncertainty. In: Henrion, M., Shachter, R., Kanal, L.N., Lemmer, J.F. (eds.) Uncertainty in Artificial Intelligence 5, pp. 29–40. Elsevier (1990)

    Google Scholar 

  19. Song, Y., Deng, Y.: Divergence measure of belief function and its application in data fusion. IEEE Access 7, 107465–107472 (2019)

    Article  Google Scholar 

  20. Song, Y., Deng, Y.: A new method to measure the divergence in evidential sensor data fusion. Int. J. Distrib. Sens. Netw. 15(4), 1550147719841295 (2019)

    Article  Google Scholar 

  21. Xiao, F.: Multi-sensor data fusion based on the belief divergence measure of evidences and the belief entropy. Inf. Fusion 46, 23–32 (2019)

    Article  Google Scholar 

  22. Yaghlane, B.B., Smets, P., Mellouli, K.: Belief function independence: I. the marginal case. Int. J. Approximate Reasoning 29(1), 47–70 (2002)

    Article  MathSciNet  Google Scholar 

  23. Yaghlane, B.B., Smets, P., Mellouli, K.: Belief function independence: II. the conditional case. Int. J. Approximate Reasoning 31(1–2), 31–75 (2002)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Faculty of Management, Prague University of Economics and Business, Jindřichův Hradec, Czechia

    Radim Jiroušek & Václav Kratochvíl

  2. Institute of Information Theory and Automation, Czech Academy of Sciences, Prague, Czechia

    Radim Jiroušek & Václav Kratochvíl

Authors
  1. Radim Jiroušek
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  2. Václav Kratochvíl
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Correspondence to Václav Kratochvíl .

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Editors and Affiliations

  1. Osaka Prefecture University, Sakai, Osaka, Japan

    Katsuhiro Honda

  2. University of Hyogo, Kobe, Japan

    Tomoe Entani

  3. Osaka Prefecture University, Sakai, Japan

    Seiki Ubukata

  4. Japan Advanced Institute of Science and Technology, Nomi, Japan

    Van-Nam Huynh

  5. Osaka University, Toyonaka, Osaka, Japan

    Masahiro Inuiguchi

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Jiroušek, R., Kratochvíl, V. (2022). Measuring Quality of Belief Function Approximations. In: Honda, K., Entani, T., Ubukata, S., Huynh, VN., Inuiguchi, M. (eds) Integrated Uncertainty in Knowledge Modelling and Decision Making. IUKM 2022. Lecture Notes in Computer Science(), vol 13199. Springer, Cham. https://doi.org/10.1007/978-3-030-98018-4_1

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  • DOI: https://doi.org/10.1007/978-3-030-98018-4_1

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