Probability-Possibility Transformations, Triangular Fuzzy Sets, and Probabilistic Inequalities

Abstract

Abstract.A possibility measure can encode a family of probability measures. This fact is the basis for a transformation of a probability distribution into a possibility distribution that generalises the notion of best interval substitute to a probability distribution with prescribed confidence. This paper describes new properties of this transformation, by relating it with the well-known probability inequalities of Bienaymé-Chebychev and Camp-Meidel. The paper also provides a justification of symmetric triangular fuzzy numbers in the spirit of such inequalities. It shows that the cuts of such a triangular fuzzy number contains the "confidence intervals" of any symmetric probability distribution with the same mode and support. This result is also the basis of a fuzzy approach to the representation of uncertainty in measurement. It consists in representing measurements by a family of nested intervals with various confidence levels. From the operational point of view, the proposed representation is compatible with the recommendations of the ISO Guide for the expression of uncertainty in physical measurement.

Formal Representations of Uncertainty

“…This constitutes another generalisation of probabilistic conditioning in the sense of equation (35). The conditional belief is then obtained by duality Bel(A || C) = 1 − P l(A || C).…”

“…The probability-possibility transforms can thus yield probabilistic inequalities. It has been shown that a symmetrical triangular possibility distribution with bounded support [a, b] is consistent with any unimodal symmetrical probability function having the same support, and contains the prediction intervals of all these probability measures (Dubois et al [35]). Moreover, it is the most specific one having these properties (it is consistent with the uniform density over [a, b]).…”

“…Indeed,it can be proved more generally that, if P (E λ ) = q, the measurable set A having the smallest measure such that P (A) = q is E λ (Dubois et al [53]; Dubois et al [35]). If p is unimodal, E λ is the interval with length L = y(λ) − s(λ) that is the most legitimate representative of the probability density p, in the sense where E λ is the interval with length L having maximal probability:…”

Analysis of the reliability of the fuzzy scale for assessing the students’ learning styles in Mathematics

“…The α-cuts or α-levels of a fuzzy set U , denoted by U α , are the intervals determined by the set of values verifying U (x) ≥ α, for α ∈ [0, 1]. Two of the most typical shapes of fuzzy sets considered in the literature are trapezoidal and triangular, where the last one is a special case of the first one (see, for instance, [8,12]). …”

Possibility Theory and Its Applications: Where Do We Stand?

“…More recently Mauris et al [81] noticed that starting from any family of nested sets around some characteristic point (the mean, the median,...), the above equation yields a possibility measure dominating P . Well-known inequalities of probability theory, such as those of Chebyshev and Camp-Meidel, can also be viewed as possibilistic approximations of probability functions.…”