A new rule to combine dependent bodies of evidence

  • Xiaoyan SuEmail author
  • Lusu Li
  • Hong Qian
  • Sankaran Mahadevan
  • Yong Deng


Dempster’s rule of combination can only be applied to independent bodies of evidence. This paper proposes a new rule to combine dependent bodies of evidence. The rule is based on the concept of joint belief distribution, and can be seen as a generalization of Dempster’s rule. When the bodies of evidence are independent, the new combination rule will be reduced into Dempster’s rule. Two examples are illustrated to show the use and effectiveness of the proposed method.


Information fusion Dempster–Shafer evidence theory Dependent evidence Generalized Dempster’s rule 



The authors greatly appreciate the reviewers’ suggestions and the editor’s encouragement. This work was partially supported by National Natural Science Foundation of China, (Grant Nos. 61503237, 61573290), “Chenguang Program” supported by Shanghai Education Development Foundation and Shanghai Municipal Education Commission, Shanghai Science and Technology Committee Key Program (Grant Nos. 18020500900, 15160500800), Shanghai Key Laboratory of Power Station Automation Technology (No. 13DZ2273800), Shanghai Education Commission Excellent Youth Project (No. ZZsdl15144).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. Antoine V, Quost B, Masson MH, Denoeux T (2014) Cevclus: evidential clustering with instance-level constraints for relational data. Soft Comput 18(7):1321–1335Google Scholar
  2. Cattaneo ME (2003) Combining belief functions issued from dependent sources. Seminar für Statistik, Eidgenössische Technische Hochschule (ETH), ZürichGoogle Scholar
  3. Cattaneo ME (2011) Belief functions combination without the assumption of independence of the information sources. Int J Approx Reason 52(3):299–315MathSciNetzbMATHGoogle Scholar
  4. Chen S, Deng Y, Wu J (2013) Fuzzy sensor fusion based on evidence theory and its application. Appl Artif Intell 27(3):235–248Google Scholar
  5. Choenni S, Blok HE, Leertouwer E (2006) Handling uncertainty and ignorance in databases: a rule to combine dependent data. In: Proceedings of the 11th international conference on database systems for advanced applications (DASFAA’06). Springer, Singapore, pp 310–324Google Scholar
  6. Coletti G, Scozzafava R (2006) Toward a general theory of conditional beliefs. Int J Intell Syst 21(3):229–259zbMATHGoogle Scholar
  7. Cuzzolin F, Gong W (2013) Belief modeling regression for pose estimation. In: Proceedings of the 16th conference on information fusion (FUSION). Istanbul, pp 1398–1405Google Scholar
  8. Dempster AP (1967) Upper and lower probabilities induced by a multivalued mapping. Ann Stat 38(2):325–339MathSciNetzbMATHGoogle Scholar
  9. Deng X, Deng Y (2018) D-AHP method with different credibility of information. Soft Comput,500-017-2993-9
  10. Deng Y, Su X, Wang D, Li Q (2010) Target recognition based on fuzzy dempster data fusion method. Def Sci J 60:525–530Google Scholar
  11. Denœux T (2008) Conjunctive and disjunctive combination of belief functions induced by nondistinct bodies of evidence. Artif Intell 172(2):234–264MathSciNetzbMATHGoogle Scholar
  12. Destercke S, Dubois D (2011) Idempotent conjunctive combination of belief functions: extending the minimum rule of possibility theory. Inf Sci 181(18):3925–3945MathSciNetzbMATHGoogle Scholar
  13. Destercke S, Dubois D, Chojnacki E (2007) Cautious conjunctive merging of belief functions. In: Symbolic and quantitative approaches to reasoning with uncertainty. Springer, Berlin, pp 332–343Google Scholar
  14. Fung R, Chong C (1985) Metaprobability and Dempster–Shafer in evidential reasoning. In: Proceedings of the 1st conference annual conference on uncertainty in artificial intelligence (UAI-85). AUAI Press, Corvallis, Oregon, pp 76–83Google Scholar
  15. Guralnik V, Mylaraswamy D, Voges H (2006) On handling dependent evidence and multiple faults in knowledge fusion for engine health management. In: Aerospace conference. IEEE, pp 9–17Google Scholar
  16. Hua Z, Gong B, Xu X (2008) A DS-AHP approach for multi-attribute decision making problem with incomplete information. Expert Syst Appl 34(3):2221–2227Google Scholar
  17. Huang S, Su X, Hu Y, Mahadevan S, Deng Y (2014) A new decision-making method by incomplete preferences based on evidence distance. Knowl Based Syst 56:264–272Google Scholar
  18. Jiang W, Zhuang M, Xie C (2017) A reliability-based method to sensor data fusion. Sensors 17(7):1575.,575 Google Scholar
  19. Kulasekere E, Premaratne K, Dewasurendra DA (2004) Conditioning and updating evidence. Int J Approx Reason 36(1):75–108MathSciNetzbMATHGoogle Scholar
  20. Liu Z, Pan Q, Dezert J (2014) Credal classification rule for uncertain data based on belief functions. Pattern Recognit 47(7):2532–2541Google Scholar
  21. Masson MH, Destercke S, Denoeux T (2016) Modelling and predicting partial orders from pairwise belief functions. Soft Comput 20(3):939–950Google Scholar
  22. Monney PA, Chan M (2007) Modelling dependence in Dempster–Shafer theory. Int J Uncertain Fuzziness Knowl Based Syst 15(1):93–114MathSciNetzbMATHGoogle Scholar
  23. Mouna C, Arnaud M, Boutheina Y (2015) Combining partially independent belief functions. Decis Support Syst 73:37–46Google Scholar
  24. Nakama T, Ruspini E (2014) Combining dependent evidential bodies that share common knowledge. Int J Approx Reason 55(9):2109–2125MathSciNetzbMATHGoogle Scholar
  25. Reformat M, Yager RR (2008) Building ensemble classifiers using belief functions and OWA operators. Soft Comput 12(6):543–558zbMATHGoogle Scholar
  26. Shafer G (1976) A mathematical theory of evidence. Princeton University Press, PrincetonzbMATHGoogle Scholar
  27. Shafer G (2016) The problem of dependent evidence. Int J Approx Reason 79(C):41–44MathSciNetzbMATHGoogle Scholar
  28. Shi F, Su X, Qian H, Yang N, Han W (2017) Research on the fusion of dependent evidence based on rank correlation coefficient. Sensors 17:2362–2377Google Scholar
  29. Smets P (1992) The concept of distinct evidence. In: Proceedings of the 4th conference on information processing and management of uncertainty in knowledge-based systems (IPMU). Palma de Mayorca, pp 789–794Google Scholar
  30. Smets P (2002) The application of the matrix calculus to belief functions. Int J Approx Reason 31(1):1–30MathSciNetzbMATHGoogle Scholar
  31. Smets P, Kennes R (1994) The transferable belief model. Artif intell 66(2):191–234MathSciNetzbMATHGoogle Scholar
  32. Su X, Deng Y, Mahadevan S, Bao Q (2012) An improved method for risk evaluation in failure modes and effects analysis of aircraft engine rotor blades. Eng Fail Anal 26:164–174Google Scholar
  33. Su X, Mahadevan S, Xu P, Deng Y (2015a) Dependence assessment in human reliability analysis using evidence theory and AHP. Risk Anal 35(7):1296–1316Google Scholar
  34. Su X, Mahadevan S, Xu P, Deng Y (2015b) Handling of dependence in Dempster–Shafer theory. Int J Intell Syst 30(4):441–467Google Scholar
  35. Su X, Mahadevan S, Han W, Deng Y (2016) Combining dependent bodies of evidence. Appl Intell 44:634–644Google Scholar
  36. Su X, Li L, Shi F, Qian H (2018) Research on the fusion of dependent evidence based on mutual information. IEEE Access 6:71839Google Scholar
  37. Voorbraak F (1991) On the justification of dempster’s rule of combination. Artif Intell 48(2):171–197MathSciNetzbMATHGoogle Scholar
  38. Wu Y, Yang J, Liu L et al (1996) On the evidence inference theory. Inf Sci 89(3):245–260MathSciNetzbMATHGoogle Scholar
  39. Xiao W, Wang Z, Wang Y (2011) Combination rule for dependent evidences. Control Decis 26(5):773–776MathSciNetGoogle Scholar
  40. Xu H, Deng Y (2018) Dependent evidence combination based on shearman coefficient and pearson coefficient. IEEE Access 6(1):11,634–11,640Google Scholar
  41. Xu H, Smets P (1994) Evidential reasoning with conditional belief functions. In: Proceedings of the 10th international conference on uncertainty in artificial intelligence, Washington, USA, pp 598–605Google Scholar
  42. Yager RR (2009) On the fusion of non-independent belief structures. Int J Gen Syst 38(5):505–531MathSciNetzbMATHGoogle Scholar
  43. Yager RR, Alajlan N (2015) Dempster-shafer belief structures for decision making under uncertainty. Knowl Based Syst 80:58–66Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Xiaoyan Su
    • 1
    Email author
  • Lusu Li
    • 1
  • Hong Qian
    • 1
  • Sankaran Mahadevan
    • 2
  • Yong Deng
    • 3
  1. 1.School of Automation EngineeringShanghai University of Electric PowerShanghaiChina
  2. 2.School of EngineeringVanderbilt UniversityNashvilleUSA
  3. 3.Department of Biostatistics, School of MedicineVanderbilt UniversityNashvilleUSA

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