On the Coherence Between Probability and Possibility Measures 1

Journal of Mathematical Psychology, 1990

This note introduces the concepts of coherent, positive coherent, and strongly coherent qualitative probability. The first interesting result is that such a qualitative probability can be extended from a given domain (not necessarily an algebra) to an arbitrary larger one. The most important result of this paper consists in proving that coherent qualitative probabilities can be represented by the Finetti's coherent previsions.

Fuzzy Sets and Systems, 1992

This paper introduces possibility integrals as fuzzy integrals w.r.t, possibility measures. It is shown that this concept can be used to unify a lot of known results in possibility theory, that up to now have been introduced on a more ad hoc basis. The integral theoretic approach allows us amongst other things to introduce product and conditional possibility measures in very much the same way as product Lebesgue measures and conditional probability measures are introduced. Furthermore, to stress the fact that the concept of order relations is basic for possibility theory, all the results are proved for a complete lattice as the evaluation set, instead of the more familiar unit interval [0, 1].

1998

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International Journal of General Systems, 1997

Information Sciences, 2004

Several authors have pointed out the relationship between consonant random sets and possibility measures. However, this relationship has only been proven for the finite case, where the inverse M€ obius of the upper probability induced by the random set simplifies the computations to a great extent. In this paper, we study the connection between both concepts for arbitrary referential spaces. We complete existing results about the lack of an implication in general with necessary and sufficient conditions for the most interesting cases.

Erkenntnis, 2005

Notre Dame Journal of Formal Logic, 2020

In recent years, some authors have proposed quantitative measures of the coherence of sets of propositions. Such probabilistic measures of coherence (PMCs) are, in general terms, functions that take as their argument a set of propositions (along with some probability distribution) and yield as their value a number that is supposed to represent the degree of coherence of the set. In this paper, I introduce a minimal constraint on PMC theories, the weak stability principle, and show that any correct, coherent, and complete PMC cannot satisfy it. As a matter of fact, the argument offered in this paper can be applied to any coherence theory that uses a priori procedures. I briefly explore some consequences of this fact.

2021

This paper reports on a geometrical investigation of de Finetti’s Dutch Book method as an operational foundation for a wide range of generalisations of probability measures, including lower probabilities, necessity measures and belief functions. Our main result identifies a number of non-limiting circumstances under which de Finetti’s coherence fails to lift from less to more general models. In particular our result shows that rich enough sets of events exist such that the coherence criteria for belief functions and lower probability collapse.

Studies in Fuzziness and Soft Computing, 2004

In probability theory the expected value of functions of random variables plays a fundamental role in defining the basic characteristic measures of probability distributions. For example, the variance, covariance and correlation of random variables can be computed as the expected value of their appropriately chosen real-valued functions. In possibility theory we can use the principle of expected value of functions on fuzzy sets to define variance, covariance and correlation of possibility distributions. Marginal probability distributions are determined from the joint one by the principle of ‘falling integrals’ and marginal possibility distributions are determined from the joint possibility distribution by the principle of ‘falling shadows’. Probability distributions can be interpreted as carriers of incomplete information [203], and possibility distributions can be interpreted as carriers of imprecise information.

Analysis, 2003

criticisms of Shogenji's coherence measure do not apply to our C. For instance, Akiba complains that if E 1 entails E 2 , then Shogenji's measure says that the degree of coherence of the set {E 1 ,E 2 } is 1/Pr(E 2 ), which he finds unintuitive, since it only depends on the 4 The fact that Shogenji's measure is always positive is a merely conventional (and therefore insignificant) violation of (1). This can be fixed simply by taking the logarithm of Shogenji's measure. The problems with Shogenji's measure discussed below are not merely conventional.