Skip to main content
SpringerLink
Log in

An intuitionistic view of the Dempster–Shafer belief structure

  • Foundations
  • Published:
Soft Computing Aims and scope Submit manuscript

Abstract

We discuss the Dempster–Shafer belief theory and describe its role in representing imprecise probabilistic information. In particular, we note its use of intervals for representing imprecise probabilities. We note in fuzzy set theory that there are two related approaches used for representing imprecise membership grades: interval-valued fuzzy sets and intuitionistic fuzzy sets. We indicate the first of these, interval-valued fuzzy sets, is in the same spirit as Dempster–Shafer representation, both use intervals. Using a relationship analogous to the type of relationship that exists between interval-valued fuzzy sets and intuitionistic fuzzy sets, we obtain from the interval-valued view of the Dempster–Shafer model an intuitionistic view of the Dempster–Shafer model. Central to this view is the use of an intuitionistic statement, pair of values, (Bel(A) Dis(A)), to convey information about the value of a variable lying in the set A. We suggest methods for combining intuitionistic statements and making inferences from these type propositions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  • Atanassov KT (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20:87–96

    Article  MathSciNet  MATH  Google Scholar 

  • Atanassov KT, Gargov G (1989) Interval valued intuitionistic fuzzy sets. Fuzzy Sets Syst 31:343–349

    Article  MathSciNet  MATH  Google Scholar 

  • Atanassov KT (2012) On intuitionistic fuzzy sets theory. Springer, Heidelberg

    Book  MATH  Google Scholar 

  • Bustince H, Burrillo P (1996) Vague sets are intuitionistic fuzzy sets. Fuzzy Sets Syst 79:403–405

    Article  MATH  Google Scholar 

  • Cornelis C, Atanassov KT, Kerre EE (2003) Intuitionistic fuzzy sets and interval valued fuzzy sets: a critical comparison. In: Proceedings of Third European Conference on Fuzzy Logic Technology (EUSFLAT) Zittau, Germany, pp 227–235

  • DeLuca A, Termini S (1972) A definition of a non-probabilistic entropy in the setting of fuzzy sets. Inf Control 20:301–312

    Article  MathSciNet  Google Scholar 

  • Dempster AP (1966) New methods of reasoning toward posterior distributions based on sample data. Ann Math Stat 37:355–374

    Article  MathSciNet  MATH  Google Scholar 

  • Dempster AP (1967) Upper and lower probabilities induced by a multi-valued mapping. Ann Math Stat 38:325–339

    Article  MathSciNet  MATH  Google Scholar 

  • Dempster AP (1968) A generalization of Bayesian inference. J R Stat Soc 30:205–247

    MathSciNet  MATH  Google Scholar 

  • Dempster AP (2008) The Dempster-Shafer calculus for statisticians. Int J Approx Reason 48:365–377

    Article  MathSciNet  MATH  Google Scholar 

  • Denoeux T (1998) Allowing imprecision in belief representation using fuzzy-valued belief structures. In: Proceedings Seventh International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, Paris, IPMU’98, vol 1, pp 48–55

  • Denoeux T (1999) Reasoning with imprecise belief structures. Int J Approx Reason 20:79–111

    Article  MathSciNet  MATH  Google Scholar 

  • Deschrijver G, Kerre EE (2003) On the relationship between some extensions of fuzzy set theory. Fuzzy Sets Syst 133:227–235

    Article  MathSciNet  MATH  Google Scholar 

  • Dymova L, Sevastjantov P (2012) The operations on intuitionistic fuzzy values in the framework of Dempster-Shafer theory. Knowl Based Syst 35:132–143

    Article  Google Scholar 

  • Dymova L, Sevastjantov P (2010) An interpretation of intuitionistic fuzzy sets in the framework of the Dempster-Shafer theory. In: Rutkowski L (ed) ICAISC 2010, Part 1, LNAI 6113, pp 66–73

  • Dymova L, Sevastjantov P (2010), An interpretation of intuitionistic fuzzy sets in terms of evidence theory decision making aspects. Knowl Based Syst 23:772–782

  • Ferson S, Kreinovich V, Ginzburg L, Sentz K, Myers DS (2002) Constructing probability boxes and Dempster-Shafer structures, Technical Report Sand 2002–4015. Sandia National Laboratories, Albuquerque

    Google Scholar 

  • Flaminio T, Godo L, Marchioni E (2013) Logics for belief functions on MV-algebras. Int J Approx Reason 54:491–512

    Article  MathSciNet  MATH  Google Scholar 

  • Karnik NN, Mendel JM (2001) Operations on type-2 fuzzy sets. Fuzzy Sets Syst 122:327–348

    Article  MathSciNet  MATH  Google Scholar 

  • Liu L, Yager RR (2008) Classic works of the Dempster-Shafer theory of belief functions: an introduction. In: Yager RR, Liu L (eds) Classic works of the Dempster-Shafer theory of belief functions. Springer, Heidelberg, pp 1–34

  • Mendel JM, Bob John RI (2002) Type-2 fuzzy sets made simple. IEEE Trans Fuzzy Syst 10:117–127

    Article  Google Scholar 

  • Mendel JM, John RI, Liu F (2006) Interval type-2 fuzzy sets made simple. IEEE Trans Fuzzy Syst 14:808–821

    Article  Google Scholar 

  • Mendel JM, Wu D (2010) Perceptual computing. Wiley, Hoboken

    Book  Google Scholar 

  • Mizumoto M, Tanaka K (1976) Some properties of fuzzy sets of type-2. Inf Control 31:312–340

    Article  MathSciNet  MATH  Google Scholar 

  • Pasi G, Atanassov K, Yager RR (2005) On intuitionistic fuzzy interpretations of elements of utility theory, Part 1. Notes Intuit Fuzzy Sets 11:62–65

  • Rickard JT, Aisbett J, Yager RR (2010) Type-2 graphical linguistic summarization and analysis. Technical Report #MII-3024 Machine Intelligence Institute. Iona College, New Rochelle

  • Shafer G (1976) A mathematical theory of evidence. Princeton University Press, Princeton

    MATH  Google Scholar 

  • Smets P (1988) Non-standard logics for automated reasoning. In: Smets P, Mamdani EH, Dubois D, Prade H (eds) Belief functions. Academic Press, London, pp 253–277

  • Smets P (1990) The combination of evidence in the transferable belief model. IEEE Trans Pattern Anal Mach Intell 12:321–344

    Article  Google Scholar 

  • Smets P, Kennes R (1994) The transferable belief model. Artif Intell 66:191–234

    Article  MathSciNet  MATH  Google Scholar 

  • Szmidt E, Kacprzyk J (2001) Entropy for intuitionistic fuzzy sets. Fuzzy Sets Syst 118:467–477

    Article  MathSciNet  MATH  Google Scholar 

  • Walley P (1982) Statistical analysis with imprecise probabilities. Chapman and Hall, London

    Google Scholar 

  • Walley P (2000) Toward unified theory of imprecise probability. Int J Approx Reason 24:125–148

    Article  MathSciNet  MATH  Google Scholar 

  • Wang ZY, Klir GJ (1992) Fuzzy measure theory. Plenum Press, New York

    Book  MATH  Google Scholar 

  • Xu Z, Yager RR (2009) Intuitionistic and inter-valued fuzzy preference relations and their measure of similarity for the evaluation of agreement within groups. Fuzzy Optim Decis Mak 8:123–139

    Article  MathSciNet  MATH  Google Scholar 

  • Yager RR, Liu L (2008) In: Dempster AP, Shafer G (eds) Classic works of the Dempster-Shafer theory of belief functions. Springer, Heidelberg

  • Yager RR (1979) On the measure of fuzziness and negation part I: membership in the unit interval. Int J Gen Syst 5:221–229

    Article  MathSciNet  MATH  Google Scholar 

  • Yager RR (1980) On the measure of fuzziness and negation part II: lattices. Inf Control 44:236–260

    Article  MathSciNet  MATH  Google Scholar 

  • Yager RR (1986) The entailment principle for Dempster-Shafer granules. Int J Intell Syst 1:247–262

    Article  MATH  Google Scholar 

  • Yager RR (1992) Decision making under Dempster-Shafer uncertainties. Int J Gen Syst 20:233–245

    Article  MATH  Google Scholar 

  • Yager RR, Kacprzyk J, Fedrizzi M (1994) Advances in the Dempster-Shafer theory of evidence. Wiley, New York

    MATH  Google Scholar 

  • Yager RR (1999) A class of fuzzy measures generated from a Dempster-Shafer belief structure. Int J Intell Syst 14:1239–1247

    Article  MATH  Google Scholar 

  • Yager RR (2001) Dempster-Shafer belief structures with interval valued focal weights. Int J Intell Syst 16:497–512

    Article  MATH  Google Scholar 

  • Yager RR (2009) Some aspects of intuitionistic fuzzy sets. Fuzzy Optim Decis Mak 8:67–90

    Article  MathSciNet  MATH  Google Scholar 

  • Yen J (1990) Generalizing the Dempster-Shafer theory to fuzzy sets. IEEE Trans Syst Man Cybern 20:559–570

    Article  MATH  Google Scholar 

  • Yen J (1992) Computing generalized belief functions for continuous fuzzy sets. Int J Approx Reason 6:1–31

    Article  MATH  Google Scholar 

  • Zadeh LA (1986) A simple view of the Dempster-Shafer theory of evidence and its implication for the rule of combination. AI Mag 7:85–90

    Google Scholar 

Download references

Acknowledgments

This work has been in part supported by ONR grant award number N00014-13-1-0626 and ARO MURI grant Number W911NF-09-1-0392. We gratefully appreciate this support.

Author information

Authors and Affiliations

  1. Machine Intelligence Institute, Iona College, New Rochelle, NY , 10801, USA

    Ronald R. Yager

Authors
  1. Ronald R. Yager
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ronald R. Yager.

Additional information

Communicated by L. G. Lacasa.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yager, R.R. An intuitionistic view of the Dempster–Shafer belief structure. Soft Comput 18, 2091–2099 (2014). https://doi.org/10.1007/s00500-014-1320-y

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-014-1320-y

Keywords

Search

Navigation