Possibility theory and statistical reasoning

Abstract

Numerical possibility distributions can encode special convex families of probability measures. The connection between possibility theory and probability theory is potentially fruitful in the scope of statistical reasoning when uncertainty due to variability of observations should be distinguished from uncertainty due to incomplete information. This paper proposes an overview of numerical possibility theory. Its aim is to show that some notions in statistics are naturally interpreted in the language of this theory. First, probabilistic inequalites (like Chebychev's) offer a natural setting for devising possibility distributions from poor probabilistic information. Moreover, likelihood functions obey the laws of possibility theory when no prior probability is available. Possibility distributions also generalize the notion of confidence or prediction intervals, shedding some light on the role of the mode of asymmetric probability densities in the derivation of maximally informative interval substitutes of probabilistic information. Finally, the simulation of fuzzy sets comes down to selecting a probabilistic representation of a possibility distribution, which coincides with the Shapley value of the corresponding consonant capacity. This selection process is in agreement with Laplace indifference principle and is closely connected with the mean interval of a fuzzy interval. It sheds light on the "defuzzification" process in fuzzy set theory and provides a natural definition of a subjective possibility distribution that sticks to the Bayesian framework of exchangeable bets. Potential applications to risk assessment are pointed out.

Coevolutionary makespan optimisation through different ranking methods for the fuzzy flexible job shop

“…Originally, in [25] the authors use the renormalisation technique (dividing the membership function µ M by its surface). However, this technique can be objected to; according to [40], it is arbitrary and the obtained probability may fail to belong to P(µ M ), the set of probability measures dominated by µ M . Here we will consider instead the probability distribution obtained from each fuzzy duration after applying the pignistic transformation obtained by considering cuts as uniformly distributed probabilities [54].…”

“…It is also the centre of the mean value of a fuzzy number as defined in [39] and the expected value of the so-called pignistic probability distribution, which is found as the centroid of the set of probabilities dominated by the possibility measure associated with X, P(Π X ) (cf. [40]). …”

A scoping inquiry into the potential contribution of Subjective Probability Theory, Dempster–Shafer Theory and Possibility Theory in accommodating degrees of belief in traveller behaviour research

“…According to Dubois (2006), the possibility degree of an event A, understood as a subset of S, is measured in Equation (16), which is calculated based on the most plausible value of x in A.…”

Application of Uncertainty Modeling Frameworks to Uncertain Isosurface Extraction

“…Instead of a single measure in probability theory, possibility theory defines a dual measure (possibility and necessity) as [7] P oss(A) = max y∈A π Y (y), N ec(A) = 1 − P oss(A c ),…”