Probability/Possibility Systems for Modeling of Random/Fuzzy Information with Parallelization Consideration



As a potential tool for handling fuzziness, possibility theory has been proposed for decades but is still far beyond mature. The fundamental reason is that we lack a clear understanding of the nature of randomness/fuzziness and then the connotation of probability/possibility. This work presented a clear definition of randomness/fuzziness and an intuitive definition of possibility as reasonable physical interpretation for existing axiomatic definition of possibility. The concepts of random/fuzzy sample spaces were introduced, upon which the axiomatic definition of possibility was properly reformulated in a structure parallel to the axiomatic definition of probability. Possibility update equation as well as possibility operators of disjunction/conjunction is discussed and justified. Though a simple rule of probability/possibility transformation is available, the whole systems of probability/possibility are found to be not coherent yet still comparable. In situations where both kinds of uncertainties are involved, fusion of one kind of uncertainty into another uncertain inference system is workable, which was further illustrated by an application example of recognizing noncooperative target using feature observations of radar cross section. Parallel computing of probability/possibility is also discussed to cope with the intensive computation challenge of practical problems of high dimension and/or with big data. We conclude that probability/possibility systems are complementary methods for handling of random/fuzzy information.


Randomness Fuzziness Possibility theory Probability–possibility transformations Bayesian inference Possibility inference Parallel computing 


  1. 1.
    Jain, A.K., Duin, R.P.W., Mao, J.: Statistical pattern recognition: a review. IEEE Trans. Pattern Anal. Mach. Intell. 22(1), 4–37 (2000)CrossRefGoogle Scholar
  2. 2.
    Cao, W., Lan, J., Li, X.R.: Conditional joint decision and estimation with application to joint tracking and classification. IEEE Trans. Syst. Man Cybern. Syst. 46(4), 459–471 (2016)CrossRefGoogle Scholar
  3. 3.
    Trken, zlem : Analysis of response surface model parameters with Bayesian approach and fuzzy approach. Int. J. Uncertainty Fuzziness Knowl. Based Syst. 24, 109–122 (2016)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Xiao, Guoqing, Li, Kenli, Zhou, Xu, Li, Keqin: Efficient monochromatic and bichromatic probabilistic reverse top-k query processing for uncertain big data. J. Comput. Syst. Sci. 89, 92–113 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Hacking, I.: The Emergence of Probability, 2nd edn. Cambridge University Press, New York (2006)CrossRefzbMATHGoogle Scholar
  6. 6.
    Zadeh, L.A.: Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst. 1(1978): 328. (Reprinted in Fuzzy Sets and Systems 100 (Supplement), 934 (1999)Google Scholar
  7. 7.
    Dubois, D., Prade, Henry: Possibility Theory and Its Applications: Where Do We Stand? Springer Handbook of Computational Intelligence. Springer, Berlin (2015)Google Scholar
  8. 8.
    Shackle, G.L.S.: Decision, Order and Time in Human Affairs, 2nd edn. Cambridge University Press, Cambridge (1961)Google Scholar
  9. 9.
    Mei, W., Shan, G.L., Li, X.R.: Simultaneous tracking and classification: a modularized scheme. IEEE Trans. Aerosp. Electron. Syst. 43(2), 581–599 (2007)CrossRefGoogle Scholar
  10. 10.
    Yager, R.R.: Conditional approach to possibility probability fusion. IEEE Trans. Fuzzy Syst. 20(1), 46–56 (2012)CrossRefGoogle Scholar
  11. 11.
    Cattaneo, M.E.G.V.: The likelihood interpretation as the foundation of fuzzy set theory. Int. J. Approx. Reason. 90, 333–340 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ferraro, M.B., Giordani, P.: Possibilistic and fuzzy clustering methods for robust analysis of non-precise data. Int. J. Approx. Reason. 88, 23–38 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Zedinia, A., Belhadjb, B.: Modeling uncertainty in monetary poverty: a possibility-based approach. Fuzzy Sets Syst. 15, 113–126 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Bellaaj, M., Elleuch, J.F., Sellami, D. et al.: An improved iris recognition system based on possibilistic modeling. In: International Conference on Advances in Mobile Computing & Multimedia. ACM (2015)Google Scholar
  15. 15.
    Raskin, V., Taylor, J.M.: Fuzziness, uncertainty, vagueness, possibility, and probability in natural language. In: IEEE Conference on Norbert Wiener in Century, pp. 1–6 (2014)Google Scholar
  16. 16.
    Coletti, G., Petturiti, D.: Finitely maxitive conditional possibilities, Bayesian-like inference, disintegrability and conglomerability. Fuzzy Sets Syst. 284, 31–55 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Jaynes, E.T.: In: Bretthorst, L. (ed.) Probability Theory: The Logic of Science. Cambridge University Press, Cambridge (2003)Google Scholar
  18. 18. (2019). Accessed 15 February 2019
  19. 19.
    Zadeh, L.A.: Outline of new approach to the analysis of complex systems and decision processes. IEEE Trans. Syst. Man Cybern. 3(1), 28–44 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20. (2019). Accessed 15 February 2019
  21. 21.
    Von Mises, R.: Probability, Statistics, and Truth (in German) (English translation, 1981: Dover Publications; 2 Revised edition. ISBN 0486242145), p. 14 (1939)Google Scholar
  22. 22.
    Liu, B.: Uncertainty Theory, 5th edn, p. 471472. Springer, Berlin (2015)Google Scholar
  23. 23.
    Dubois, D., Moral, S., Prade, H.: A semantics for possibility theory based on likelihoods. J. Math. Anal. Appl. 205, 359–380 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Dubois, D., Foulloy, L., et al.: Probability-possibility transformations, triangular fuzzy sets, and probabilistic inequalities. Reliable Comput. 10, 273–297 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Jina, L., Kalina, M., Mesiar, R.: Characterizations of the possibility-probability transformations and some applications. Inf. Sci. 477, 281–290 (2019)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Ferson, S.: Bayesian methods in risk assessment, Technical Report (2005).
  27. 27.
    Mei, W.: Bridging probability and possibility via Bayesian theorem. Int. J. Uncertainty Fuzziness Knowl. Based Syst. 4, 615–626 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Coletti, G., Scozzafava, R.: Conditional probability, fuzzy sets, and possibility: a unifying view. Fuzzy Sets Syst. 144(1), 227–249 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Oussalah, M.: On the probability/possibility transformations: a comparative analysis. J. General Syst. 29(5), 671–718 (2000)CrossRefzbMATHGoogle Scholar
  30. 30.
    Klir, G.J.: Information-preserving probability-possibility transformations: recent developments. Fuzzy Logic 417–428 (1993)Google Scholar
  31. 31.
    Mouchaweh, M.S., Billaudel, P.: Variable probability-possibility transformation for the diagnosis by pattern recognition. Int. J. Comput. Intell. Theory Pract. 1, 9–21 (2006)Google Scholar
  32. 32.
    Dhar, M.: A revisit to probability–possibility consistency principles. Int. J. Intell. Syst. Appl. 5(4), 90–99 (2013)Google Scholar
  33. 33.
    Dubois, D., Prade, H.: Fuzzy Sets and Systems: Theory and Applications. Academic Press, New York (1980)zbMATHGoogle Scholar
  34. 34.
    Delgado, M., Moral, S.: On the concept of possibility–probability consistency. Fuzzy Sets Syst. 21, 311–318 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Lapointe, S., Bobe, B.: Revision of possibility distributions: a Bayesian inference pattern. Fuzzy Sets Syst. 116(2), 119–140 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Kontoghiorghes, E.J.: Handbook of Parallel Computing and Statistics, Technometrics. Chapman & Hall/CRC, Boca Raton (2008). Google Scholar
  37. 37.
    Zhu, J., Chen, J., Hu, W., et al.: Big learning with Bayesian methods. Nat. Sci. Rev. 4, 627–651 (2017)CrossRefGoogle Scholar
  38. 38.
    Guo, G.: Parallel statistical computing for statistical inference. J. Stat. Theory Pract. 6(3), 536–565 (2012)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Flynn, M.J.: Some computer organizations and their effectiveness. IEEE Trans. Comput. C–21, 948–960 (1972)CrossRefzbMATHGoogle Scholar
  40. 40.
    Schmidberger, M.: Parallel computing for biological data, Dissertation, University of Munich, Germany (2009)Google Scholar
  41. 41.
  42. 42.
    Patterson, D.A.: The parallel computing landscape: a Berkeley view. In: ACM/IEEE International Symposium on Low Power Electronics & Design (2007)Google Scholar
  43. 43.
    Lukasik, S.: Parallel computing of kernel density estimates with MPI. Lect. Notes Comput. Sci. 4489, 726–734 (2007)CrossRefGoogle Scholar
  44. 44.
    Garcia, E., Hausotte, T.: The parallel bayesian toolbox for high-performance Bayesian filtering in metrology. Meas. Sci. Rev. 13(6), 315–321 (2013)CrossRefGoogle Scholar
  45. 45.
    Chen, J., Li, K., Tang, Z., et al.: A parallel random forest algorithm for big data in Spark cloud computing environment. IEEE Trans. Parallel Distrib. Syst. 28(4), 919–933 (2017)CrossRefGoogle Scholar
  46. 46.
    Chen, Y., Li, K., Yang, W. et al.: Performance-aware model for sparse matrix-matrix multiplication on the sunway taihulight supercomputer. IEEE Trans. Parallel Distrib. Syst. (2018).
  47. 47.
    Guo, P., Zhu, B., Niu, H., et al.: Fast genomic prediction of breeding values using parallel Markov chain Monte Carlo with convergence diagnosis. BMC Bioinf. 19(1), 3 (2018)CrossRefGoogle Scholar
  48. 48.
    Mei, W.: Probability/possibility systems for treatment of random/fuzzy knowledge. In: International Conference on Natural Computation, Fuzzy Systems and Knowledge Discovery, pp. 573–579 (2018)Google Scholar

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© Taiwan Fuzzy Systems Association 2019

Authors and Affiliations

  1. 1.Army Engineering UniversityShijiazhuangPeople’s Republic of China

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