Basic Issues of Computing with Granular Probabilities
Imprecision of probabilities is needed to reflect the amount of information on which they are based. The precision should increase with the amount of information available.
Total ignorance can be properly modeled by vacuous probabilities, which are maximally imprecise (i.e., each covers the whole range [0, 1]), but not by any precise probabilities.
Imprecise probabilities are generally easier to assess and elicit than precise ones.
We may be unable to assess probabilities precisely in practice, even if that is possible in principle, because we lack the time and computational ability.
A precise probability model that is defined on some class of events determines only imprecise probabilities for events outside the class.
When several sources of information (sensors, experts, individuals in a group decision) are combined, the extent to which they are consistent can be reflected in the precision of the combined model.
KeywordsFuzzy Number Bayesian Inference Triangular Membership Function Additivity Axiom Fuzzy Interval
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- Kaufmann, A. and M.M.Gupta , Introduction to Fuzzy Arithmetic. Van Nostrand, New York.Google Scholar
- Klir, G. J. [ 1997 ], “The role of fuzzy arithmetic in engineering.” In: Ayyub, B. M., ed., Uncertainty Analysis in Engineering and the Science. Kluwer, Boston.Google Scholar
- Klir, G. J. and B. Yuan [ 1995 ], Fuzzy Sets and Fuzzy Logic: Theory and Applications. Prentice Hall, Upper Saddle River, NJ.Google Scholar
- Pan, Y. [ 1997a ], “Revised hierarchical analysis method based on crisp and fuzzy entries and its application to assessment of prior probability distributions.” Inter. J. of General Systems, 26 (1–2), pp. 1110–131.Google Scholar
- Pan, Y. [ 1997b ], Calculus of Fuzzy Probabilities and Its Applications. PhD Dissertation in Systems Science, Binghamton University-SUNY.Google Scholar
- Pan, Y. and G. J. Klir , “Bayesian inference based on interval-valued prior distributions and likelihoods.” J. of Intelligent & Fuzzy Systems,5(3).Google Scholar
- Zadeh, L. A. , “The concept of a linguistic variable and its application to approximate reasoning I, II, III.” Information Sciences, 8, pp. 199–251, 301–357; 9, pp. 43–80.Google Scholar