# A New Perspective of Bayes Formula Based on D–S Theory in Interval Intuitionistic Fuzzy Environment and Its Applications

- 56 Downloads

## Abstract

Uncertainty inference, which is used to infer the reasons corresponding to the results, has permeated through various fields, such as medicine, risk assessments, and marine management. Tackling ambiguity under fuzzy environment by using the Bayes and Dempster–Shafer theories has recently received special attention. In actual cases and experiments, we can estimate probabilities of occurrence and non-occurrence of events, which are expressed by intuitionistic fuzzy numbers (IFNs) or interval-valued intuitionistic fuzzy numbers (IVIFNs). To accomplish the multi-source uncertain inference in a fuzzy environment, this paper proposes the standard forms of Bayes formula under the intuitionistic fuzzy environment and the interval-valued intuitionistic fuzzy environment to solve grey models and decision-making problems with incomplete information. Firstly, we give some basic properties of IFN-probability and define its conditional probability based on the basic probability assignment of Dempster–Shafer theory. Subsequently, we also propose the independence of events, multiplication rule for the conditional probability of IFNs, and the law of total probability and the Bayes formula of IFNs. Due to the complexity and uncertainty of objective things, the indeterminacy degrees are often difficult to be expressed with crisp numbers. To address the challenge, interval values are used to describe the indeterminacy degrees instead of crisp numbers. Secondly, we give some basic properties of IVIFN-probability and define its conditional probability based on the IFN-probability theory. Afterward, we further propose the independence of events, multiplication rule for the conditional probability of IVIFNs, and the law of total probability and the Bayes formula of IVIFNs. From the angle of modeling and complexity, our approaches are more convenient and feasible than the interval probability method. Meanwhile, the practical applications show the feasibility of the proposed methods.

## Keywords

Uncertainty inference Independence Conditional probability Law of total probability Bayes formula Intuitionistic fuzzy environment Interval-valued intuitionistic fuzzy environment## Notes

### Acknowledgements

The work was supported in part by the China National Natural Science Foundation (Nos. 71771155, 71571123), and the Scholarship from China Scholarship Council (No. 201706240012).

## References

- 1.Zadeh, L.A.: Fuzzy sets. Inf. Control
**8**(3), 338–353 (1965)zbMATHGoogle Scholar - 2.Atanassov, K.T.: Intuitionistic fuzzy-sets. Fuzzy Sets Syst.
**20**(1), 87–96 (1986)MathSciNetzbMATHGoogle Scholar - 3.Xu, Z.S., Yager, R.R.: Some geometric aggregation operators based on intuitionistic fuzzy sets. Int. J. Gener. Syst.
**35**(4), 417–433 (2006)MathSciNetzbMATHGoogle Scholar - 4.Xu, Z.S.: Intuitionistic fuzzy aggregation operators. IEEE Trans. Fuzzy Syst.
**15**(6), 1179–1187 (2007)Google Scholar - 5.Atanassov, K.T., Gargov, G.: Interval valued intuitionistic fuzzy-sets. Fuzzy Sets Syst.
**31**(3), 343–349 (1989)MathSciNetzbMATHGoogle Scholar - 6.Atanassov, K.T.: Operators over interval valued intuitionistic fuzzy sets. Fuzzy Sets Syst.
**64**(2), 159–174 (1994)MathSciNetzbMATHGoogle Scholar - 7.Xu, Z.S.: Methods for aggregating interval-valued intuitionistic fuzzy information and their application to decision making. Control Decis.
**22**(2), 215–219 (2007)MathSciNetGoogle Scholar - 8.Zhou, W., Xu, Z.S., Chen, M.H.: Preference relations based on hesitant-intuitionistic fuzzy information and their application in group decision making. Comput. Ind. Eng.
**87**, 163–175 (2015)Google Scholar - 9.Xia, M.M.: Interval-valued intuitionistic fuzzy matrix games based on Archimedean t-conorm and t-norm. Int. J. Gener. Syst.
**47**(3), 278–293 (2018)MathSciNetGoogle Scholar - 10.Tao, Z.F., Chen, H.Y., Zhou, L.G.: A generalized multiple attributes group decision making approach based on intuitionistic fuzzy sets. Int. J. Fuzzy Syst.
**16**(2), 184–195 (2014)MathSciNetGoogle Scholar - 11.Bharati, S.K., Singh, S.R.: Transportation problem under interval-valued intuitionistic fuzzy environment. Int. J. Fuzzy Syst.
**20**(5), 1511–1522 (2018)Google Scholar - 12.Zadeh, L.A.: Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst.
**1**(1), 3–28 (1978)MathSciNetzbMATHGoogle Scholar - 13.Zadeh, L.A.: Probability measures of fuzzy events. J. Math. Anal. Appl.
**23**(2), 421–427 (1968)MathSciNetzbMATHGoogle Scholar - 14.Carlsson, C., Fuller, R.: On possibilistic mean value and variance of fuzzy numbers. Fuzzy Sets Syst.
**122**(2), 315–326 (2001)MathSciNetzbMATHGoogle Scholar - 15.Fuller, R., Majlender, N.: On weighted possibilistic mean and variance of fuzzy numbers. Fuzzy Sets Syst.
**136**(3), 363–374 (2003)MathSciNetzbMATHGoogle Scholar - 16.Grzegorzewski, P. and Mrówka, E.: Probability of intuitionistic fuzzy events. In: Soft Methods in Probability, Statistics and Data Analysis, pp. 105–115 (2002)Google Scholar
- 17.Riečan, B.: Representation of Probabilities on IFS Events, vol. 26, pp. 243–248. Springer, Berlin (2004)zbMATHGoogle Scholar
- 18.Ciungu, L.C., Riečan, B.: Representation theorem for probabilities on IFS-events. Inf. Sci.
**180**(5), 793–798 (2010)MathSciNetzbMATHGoogle Scholar - 19.Grzegorzewski, P.: On some basic concepts in probability of IF-events. Inf. Sci.
**232**, 411–418 (2013)MathSciNetzbMATHGoogle Scholar - 20.Snoek, J., Larochelle, H., Adams, R.P.: Practical Bayesian optimization of machine learning algorithms. In: International Conference on Neural Information Processing Systems, vol. 4, pp. 2951–2959 (2012)Google Scholar
- 21.Erte, P., Zhu, H.: Non-parametric Bayesian learning with deep learning structure and its applications in wireless networks. In: Signal Information Processing, pp. 1233–1237 (2014)Google Scholar
- 22.Li, B., Han, T., Kang, F.Y.: Fault diagnosis expert system of semiconductor manufacturing equipment using a Bayesian network. Int. J. Comput. Integr. Manuf.
**26**(12), 1161–1171 (2013)Google Scholar - 23.Wang, X.L., Zang, M., Xiao, G.H.: Epigenetic change detection and pattern recognition via Bayesian hierarchical hidden Markov models. Stat. Med.
**32**(13), 2292–2307 (2013)MathSciNetGoogle Scholar - 24.Ellison, A.M.: Bayesian inference in ecology. Ecol. Lett.
**7**(6), 509–520 (2004)Google Scholar - 25.Fong, Y., Rue, H., Wakefield, J.: Bayesian inference for generalized linear mixed models. Biostatistics
**11**(3), 397–412 (2010)Google Scholar - 26.Sun, H., Betti, R.: A hybrid optimization algorithm with Bayesian inference for probabilistic model updating. Comput. Aided Civ. Inf.
**30**(8), 602–619 (2015)Google Scholar - 27.Khakzad, N., Khan, F., Amyotte, P.: Safety analysis in process facilities: comparison of fault tree and Bayesian network approaches. Reliab. Eng. Syst. Saf.
**96**(8), 925–932 (2011)Google Scholar - 28.Shahriar, B., Swersky, K., Wang, Z., Adams, R.P., Freitas, N.D.: Taking the human out of the loop: a review of Bayesian optimization. Proc. IEEE
**104**(1), 148–175 (2016)Google Scholar - 29.Jia, Y.H., Kwong, S., Wu, W.H.: Sparse Bayesian learning-based Kernel Poisson regression. IEEE Trans. Cybern.
**99**, 1–13 (2017)Google Scholar - 30.Junttila, V., Laine, M.: Bayesian principal component regression model with spatial effects for forest inventory variables under small field sample size. Remote Sens. Environ.
**192**, 45–57 (2017)Google Scholar - 31.Ma, F., Chen, Y.W., Yan, X.P.: A novel marine radar targets extraction approach based on sequential images and Bayesian network. Ocean Eng.
**120**, 64–77 (2016)Google Scholar - 32.Mohammadfam, I., Ghasemi, F., Kalatpour, O.: Constructing a Bayesian network model for improving safety behavior of employees at workplaces. Appl. Ergon.
**58**, 35–47 (2017)Google Scholar - 33.Malagrino, L.S., Roman, N.T., Monteiro, A.M.: Forecasting stock market index daily direction: a Bayesian network approach. Expert Syst. Appl.
**105**, 11–22 (2018)Google Scholar - 34.Pouymayou, B., Riesterer, O., Guckenberger, N., Unkelbach, J.: A Bayesian network model of lymphatic tumor progression for personalized elective CTV definition. Med. Phys.
**45**(6), E150–E150 (2018)Google Scholar - 35.Castelletti, A., Soncini-Sessa, R.: Bayesian networks and participatory modelling in water resource management. Environ. Modell. Softw.
**22**(8), 1075–1088 (2007)Google Scholar - 36.Zwirglmaier, K., Straub, D., Groth, K.M.: Capturing cognitive causal paths in human reliability analysis with Bayesian network models. Reliab. Eng. Syst. Saf.
**158**, 117–129 (2017)Google Scholar - 37.Dempster, A.P.: Upper and lower probabilities induced by a multivalued mapping. Ann. Math. Stat.
**38**(2), 325–339 (1967)MathSciNetzbMATHGoogle Scholar - 38.Shafer, G.: A Mathematical Theory of Evidence. Princeton University Press, New Jersey (1976)zbMATHGoogle Scholar
- 39.Xu, D.L., Yang, J.B., Wang, Y.M.: The evidential reasoning approach for multi-attribute decision analysis under interval uncertainty. Eur. J. Oper. Res.
**174**(3), 1914–1943 (2006)zbMATHGoogle Scholar - 40.Zhou, H., Wang, J.Q., Zhang, H.Y., Chen, X.H.: Linguistic hesitant fuzzy multi-criteria decision-making method based on evidential reasoning. Int. J. Syst. Sci.
**47**(2), 314–327 (2016)MathSciNetzbMATHGoogle Scholar - 41.Du, Y.W., Wang, Y.M.: Evidence combination rule with contrary support in the evidential reasoning approach. Expert Syst. Appl.
**88**, 193–204 (2017)Google Scholar - 42.Ye, J.M., Xu, Z.S., Gou, X.J.: Virtual linguistic trust degree and BUM for evidential reasoning for the emergency response assessment of railway stations emergency. Technical report (2018)Google Scholar
- 43.Pratama, V.A., Natalia, F.: A Dempster–Shafer approach to an expert system design in diagnosis of febrile disease. In: International Conference on New Media Studies, pp. 62–68 (2017)Google Scholar
- 44.Sun, L., Wang, Y.Z.: A multi-attribute fusion approach extending Dempster–Shafer theory for combinatorial-type evidences. Expert Syst. Appl.
**96**, 218–229 (2018)Google Scholar - 45.Xing, Q.H., Liu, F.X.: Method of determining membership and nonmembership function in intuitionistic fuzzy sets. Control and Decis.
**24**(3), 393–397 (2009)MathSciNetGoogle Scholar - 46.Tessem, B.: Interval probability propagation. Int. J. Approx. Reason.
**7**(3–4), 95–120 (1992)MathSciNetzbMATHGoogle Scholar - 47.Weichselberger, K.: The theory of interval-probability as a unifying concept for uncertainty. Int. J. Approx. Reason.
**24**(2–3), 149–170 (2000)MathSciNetzbMATHGoogle Scholar - 48.Guo, P.J., Tanaka, H.: Decision making with interval probabilities. Eur. J. Oper. Res.
**203**(2), 444–454 (2010)MathSciNetzbMATHGoogle Scholar - 49.Senguptaand, A., Pal, T.K.: On comparing interval numbers. Eur. J. Oper. Res.
**127**(1), 28–43 (2000)MathSciNetGoogle Scholar - 50.Zhou, L.G., Chen, H.Y., Gil-Lafuente, A.M.: Uncertain generalized aggregation operators. Expert Syst. Appl.
**39**(1), 1105–1117 (2012)Google Scholar - 51.Dempster, A.P., Weisberg, H.: A generalization of Bayesian inference. J. R. Stat. Soc. B
**30**(2), 205–247 (1968)MathSciNetzbMATHGoogle Scholar - 52.Shafer, G.: Belief functions and possibility measures. Anal. of Fuzzy Inf.
**1**, 51–84 (1987)MathSciNetzbMATHGoogle Scholar - 53.Hong, D.H., Choi, C.H.: Multicriteria fuzzy decision-making problems based on vague set theory. Fuzzy Sets Syst.
**114**(1), 103–113 (2000)zbMATHGoogle Scholar