Basic Issues of Computing with Granular Probabilities

  • George J. Klir
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 95)


It has increasingly been recognized that practical applicability of classical probability theory to some problems of current interest, particularly those dependent on human judgement, is severely restricted. This restriction is a result of the additivity axiom of classical probability theory. Due to this axiom, elementary events are required to be pairwise disjoint and the probability of each is required to be expressed precisely by a real number in the unit interval [0, 1]. The requirement of disjoint events makes mathematically good sense, but it becomes problematic whenever we leave the world of mathematics. As is well known, there is no method of scientific measurement that is free from error. As a consequence, observations in close neighborhoods of the sharp boundaries between events are unreliable and should be properly discounted. This results in a violation of the additivity axiom and, by implication, in a violation of the precision requirement. Hence, we need to deal with imprecise probabilities. This need is even more pronounced when instead of measurements we are dependent on assessments based on subjective human judgement. There are other convincing arguments for imprecise probabilities, including, for example, the following:
  • 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.


Fuzzy Number Bayesian Inference Triangular Membership Function Additivity Axiom Fuzzy Interval 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • George J. Klir
    • 1
  1. 1.Center for Intelligent Systems and Dept. of Systems Science & Industrial Eng. Binghamton UniversitySUNYBinghamtonUSA

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