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\(\textbf{r}\)-ERBFN: An Extension of the Evidential RBFN Accounting for the Dependence Between Positive and Negative Evidence

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Scalable Uncertainty Management (SUM 2024)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 15350))

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Abstract

Recently, it was shown that a radial basis function network (RBFN) with a softmax output layer amounts to pooling by Dempster’s rule positive and negative evidence for each class, and approximating the resulting belief function by a probability distribution using the plausibility transform. This so-called latent belief function offers a richer uncertainty quantification than the probabilistic output of the RBFN. In this paper, we show that there exists actually a set of latent belief functions for a RBFN. This set is obtained by considering all possible dependence structures, which are described by correlations, between the positive and negative evidence for each class. Furthermore, we show that performance can be enhanced by optimizing the correlations brought to light.

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Notes

  1. 1.

    Example 1 is based on the probabilistic dependence structure considered in [23, Example 1].

  2. 2.

    We focus for short on the case \(K>2\) in this section, but our developments also hold for \(K=2\).

  3. 3.

    Enforcing at least two training examples of the given class per cluster.

  4. 4.

    Pima is available from the R package MASS [26]. Ionosphere, Glass and Vowel are available from the UCI ML repository https://archive.ics.uci.edu. For Vowel, we considered only the first six classes, as in [11].

  5. 5.

    P-value obtained for the Glass dataset.

References

  1. Bradley, P.S., Bennett, K.P., Demiriz, A.: Constrained k-means clustering. Technical report. MSR-TR-2000-65, Microsoft Research, Redmond (2000). www.microsoft.com/en-us/research/wp-content/uploads/2016/02/tr-2000-65.pdf

  2. Cobb, B.R., Shenoy, P.P.: On the plausibility transformation method for translating belief function models to probability models. Int. J. Approx. Reason. 41(3), 314–330 (2006)

    Article  MathSciNet  Google Scholar 

  3. Dempster, A.P.: Upper and lower probabilities induced by a multivalued mapping. Ann. Math. Stat. 38, 325–339 (1967)

    Article  MathSciNet  Google Scholar 

  4. Denœux, T.: A \(k\)-nearest neighbor classification rule based on Dempster-Shafer theory. IEEE Trans. Syst. Man Cybern. 25(5), 804–213 (1995)

    Article  Google Scholar 

  5. Denœux, T.: A neural network classifier based on Dempster-Shafer theory. IEEE Trans. Syst. Man Cybern. - Part A 30(2), 131–150 (2000)

    Google Scholar 

  6. Denœux, T.: Quantifying predictive uncertainty using belief functions: different approaches and practical construction. In: Kreinovich, V., Sriboonchitta, S., Chakpitak, N. (eds.) TES 2018. SCI, vol. 753, pp. 157–176. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-70942-0_8

    Chapter  Google Scholar 

  7. Denœux, T.: Logistic regression, neural networks and Dempster-Shafer theory: a new perspective. Knowl.-Based Syst. 176, 54–67 (2019)

    Article  Google Scholar 

  8. Denœux, T.: Belief functions induced by random fuzzy sets: a general framework for representing uncertain and fuzzy evidence. Fuzzy Sets Syst. 424, 63–91 (2021)

    Article  MathSciNet  Google Scholar 

  9. Denœux, T.: Théorie des fonctions de croyance et apprentissage automatique, (2022). journée Apprentissage automatique multimodal et fusion d’informations (2ème édition), GdR ISIS, virtual, 19th January 2022

    Google Scholar 

  10. Denœux, T.: Quantifying prediction uncertainty in regression using random fuzzy sets: the ENNreg model. IEEE Trans. Fuzzy Syst. 31(10), 3690–3699 (2023)

    Article  Google Scholar 

  11. Denœux, T.: Uncertainty quantification in logistic regression using random fuzzy sets and belief functions. Int. J. Approx. Reason. 168, 109159 (2024)

    Article  MathSciNet  Google Scholar 

  12. Denœux, T.: Combination of dependent and partially reliable Gaussian random fuzzy numbers. Inf. Sci. 681, 121208 (2024)

    Google Scholar 

  13. Dubois, D., Faux, F., Prade, H.: Prejudice in uncertain information merging: pushing the fusion paradigm of evidence theory further. Int. J. Approx. Reason. 121, 1–22 (2020)

    Article  MathSciNet  Google Scholar 

  14. Ferson, S., et al.: Dependence in probabilistic modeling, Dempster-Shafer theory, and probability bounds analysis. Technical report. SAND2004-3072, Sandia Nat. Lab., Albuquerque, New Mexico (2004)

    Google Scholar 

  15. Fréchet, M.: Généralisations du théorème des probabilités totales. Fundam. Math. 25, 379–387 (1935)

    Google Scholar 

  16. Huang, L., Ruan, S., Decazes, P., Denœux, T.: Lymphoma segmentation from 3D PET-CT images using a deep evidential network. Int. J. Approx. Reason. 149, 39–60 (2022)

    Google Scholar 

  17. Hüllermeier, E., Waegeman, W.: Aleatoric and epistemic uncertainty in machine learning: an introduction to concepts and methods. Mach. Learn. 110, 457–506 (2021)

    Google Scholar 

  18. Klir, G.J.: Uncertainty and Information: Foundations of Generalized Information Theory. Wiley-IEEE Press, Hoboken (2005)

    Google Scholar 

  19. Ronneberger, O., Fischer, P., Brox, T.: U-Net: convolutional networks for biomedical image segmentation. In: Navab, N., Hornegger, J., Wells, W.M., Frangi, A.F. (eds.) MICCAI 2015. LNCS, vol. 9351, pp. 234–241. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-24574-4_28

  20. Schwenker, F., Kestler, H.A., Palm, G.: Three learning phases for radial-basis-function networks. Neural Netw. 14(4), 439–458 (2001)

    Article  Google Scholar 

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

    Google Scholar 

  22. Shafer, G.: Probability judgment in artificial intelligence. In: Kanal, L.N., Lemmer, J.F. (eds.) Uncertainty in Artificial Intelligence, Machine Intelligence and Pattern Recognition, vol. 4, pp. 127–135. North-Holland (1986)

    Google Scholar 

  23. Shenoy, P.: On distinct belief functions in the Dempster-Shafer theory. In: Miranda, E., Montes, I., Quaeghebeur, E., Vantaggi, B. (eds.) Proceedings of the Thirteenth International Symposium on Imprecise Probability: Theories and Applications, vol. 215, pp. 426–437. PMLR (2023)

    Google Scholar 

  24. Smets, P.: Belief functions: the disjunctive rule of combination and the generalized Bayesian theorem. Int. J. Approx. Reason. 9(1), 1–35 (1993)

    Article  MathSciNet  Google Scholar 

  25. Tong, Z., Xu, P., Denœux, T.: An evidential classifier based on Dempster-Shafer theory and deep learning. Neurocomputing 450, 275–293 (2021)

    Article  Google Scholar 

  26. Venables, W.N., Ripley, B.D.: Modern Applied Statistics with S, fourth edn. Springer, New York (2002). https://doi.org/10.1007/978-0-387-21706-2

  27. Voorbraak, F.: A computationally efficient approximation of Dempster-Shafer theory. Int. J. Man-Mach. Stud. 30(5), 525–536 (1989)

    Article  Google Scholar 

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Acknowledgments

Serigne Diène’s PhD work is funded by the Hauts-de-France region and Artois University.

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Authors and Affiliations

  1. Univ. Artois, EA 3926 LGI2A, 62400, Béthune, France

    Frédéric Pichon, Serigne Diène, Sébastien Ramel & David Mercier

  2. Université de technologie de Compiègne, CNRS, Heudiasyc, Compiègne, France

    Thierry Denœux

  3. Institut universitaire de France, Paris, France

    Thierry Denœux

Authors
  1. Frédéric Pichon
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  2. Serigne Diène
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  3. Thierry Denœux
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  4. Sébastien Ramel
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  5. David Mercier
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Corresponding author

Correspondence to Frédéric Pichon .

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Editors and Affiliations

  1. Université de Technologie de Compiègne, Compiègne, France

    Sébastien Destercke

  2. Artificial Intelligence Research Institute, Barcelona, Spain

    Maria Vanina Martinez

  3. University of Palermo, Palermo, Italy

    Giuseppe Sanfilippo

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Pichon, F., Diène, S., Denœux, T., Ramel, S., Mercier, D. (2025). \(\textbf{r}\)-ERBFN: An Extension of the Evidential RBFN Accounting for the Dependence Between Positive and Negative Evidence. In: Destercke, S., Martinez, M.V., Sanfilippo, G. (eds) Scalable Uncertainty Management. SUM 2024. Lecture Notes in Computer Science(), vol 15350. Springer, Cham. https://doi.org/10.1007/978-3-031-76235-2_26

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  • DOI: https://doi.org/10.1007/978-3-031-76235-2_26

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