Estimations of expectedness and potential surprise in possibility theory

This note investigates how various ideas of 'expectedness' can be captured in the framework of possibility theory. Particularly, we are interested in trying to introduce estimates of the kind of lack of surprise expressed by people when saying 'I would not be surprised that...' before an event takes place, or by saying 'I knew it' after its realization. In possibility theory, a possibility distribution is supposed to model the relative levels of mutually exclusive alternatives in a set, or equivalently, the alternatives are assumed to be rank-ordered according to their level of possibility to take place. Four basic set-functions associated with a possibility distribution, including standard possibility and necessity measures, are discussed from the point of view of what they estimate when applied to potential events. Extensions of these estimates based on the notions of Q-projection or OWA operators are proposed when only significant parts of the possibility distribution are retained in the evaluation. The case of partially-known possibility distributions is also considered. Some potential applications are outlined

A Functional Representation of Potential Surprise Ordering

In the history of economic thought, Shackle was one of the representative critics about probability based economic theory. Specifically, he constructed his own concept of subjective uncertainty called potential surprise to replace probability. In 1980s, the potential surprise is axiomatized by Katzner as Kolmogorov-styled measure defined on the -field over the set of possible states. In this paper, potential surprise function is reconstructed as the functional representation of potential surprise ordering on the space of hypotheses about future called monad

Estimations of expectedness and potential surprise in possibility theory

This paper is a revised version of a short note which appeared in the Proceedings of the North-American Fuzzy Information Processing Society Conference (NAFIPS'92), held at Puerto Vallarta, Mexico on December 15-17, 1992, (NASA Conference Publication 10112) Vol. 1 pp.7-13.International audienceThis note investigates how various ideas of "expectedness" can be captured in the framework of possibility theory. Particularly, we are interested in trying to introduce estimates of the kind of lack of surprise expressed by people when saying "I would not be surprised that…" before an event takes place, or by saying "I knew it" after its realization. In possibility theory, a possibility distribution is supposed to model the relative levels of possibility of mutually exclusive alternatives in a set, or equivalently, the alternatives are assumed to be rank-ordered according to their level of possibility to take place. Four basic set-functions associated with a possibility distribution, including standard possibility and necessity measures, are discussed from the point of view of what they estimate when applied to potential events. Extensions of these estimates based on the notions of Q-projection or OWA operators are proposed when only significant parts of the possibility distribution are retained in the evaluation. The case of partially-known possibility distributions is also considered. Some potential applications are outlined

Estimations Of Expectedness And Potential Surprise In Possibility Theory

This paper is a revised version of a short note which appeared in the Proceedings of the NorthAmerican Fuzzy Information Processing Society Conference (NAFIPS'92), held at Puerto Vallarta, Mexico on December 15-17, 1992, (NASA Conference Publication 10112) Vol.1 pp.7-13. distribution. Such a distribution assesses the level of possibility of each possible value of a considered (single-valued) variable x, i.e., the elements of the domain of the variable x are rank-ordered according to their relative possibility on the scale [0,1]. Then a possibility measure Õ is associated with the distribution, and Õ(A) estimates the consistency of the available knowledge with the statement "A is true" (short for "x is in A is true"). A dual measure of necessity N estimates the certainty of A as the impossibility of "non A", namely N(A) = Impos(non A) = 1 -- Õ(non A). Recently, Gärdenfors and Makinson [3] have called 'expectation' a function obeying the characteristic property of a necessity measure, namely, N(AÇB) = min (N(A), N(B)), but which may not be valued on [0,1]. Besides, N(non A), the certainty that A is false, can be interpreted as a degree of surprise S(A) = N(non A) = Impos(A) that A is true. This corresponds exactly to the view developed by the English economist Shackle [6] who worked out a non-probabilistic model of expectation, before the introduction of possibility theory. However this notion of surprise where Õ(A) = 1 -- S(A) does not seem to correspond exactly to the intended meaning of a sentence such that "I would not be surprised that A is true", which rather expresses that "A true" is more than just possible (even with a high degree), and is not far to be somewhat certain; what is stated is a very strong kind of possibility. In this note we investigate what estim..