A counterpart to von Neumann and Morgenstern' expected utility theory is proposed in the framework of possibility theory. The existence of a utility function, representing a preference ordering among possibility distributions (on the consequences of decision-maker's actions) that satisfies a series of axioms pertaining to decision-maker's behavior, is established. The obtained utility is a generalization of Wald's criterion, which is recovered in case of total ignorance; when ignorance is only partial, the utility takes into account the fact that some situations are more plausible than others. Mathematically, the qualitative utility is nothing but the necessity measure of a fuzzy event in the sense of possibility theory (a so-called Sugeno integral). The possibilistic representation of uncertainty, which only requires a linearly ordered scale, is qualitative in nature. Only max, min and order-reversing operations are used on the scale. The axioms express a risk-averse behavior of the d...

Pathfinder is an expert system that assists surgical pathologists with the diagnosis of lymph-node diseases. The program is one of a growing number of normative expert systems that use probability and decision theory to acquire, represent, manipulate, and explain uncertain medical knowledge. In this article, we describe Pathfinder and our research in uncertain-reasoning paradigms that was stimulated by the development of the program. We discuss limitations with early decision-theoretic methods for reasoning under uncertainty and our initial attempts to use non-decision-theoretic methods. Then, we describe experimental and theoretical results that directed us to return to reasoning methods based in probability and decision theory.

This paper is meant to survey the literature pertaining to this debate, and to try to overcome misunderstandings and to supply access to many basic references that have addressed the "probability versus fuzzy set" challenge. This problem has not a single facet, as will be claimed here. Moreover it seems that a lot of controversies might have been avoided if protagonists had been patient enough to build a common language and to share their scientific backgrounds. The main points made here are as follows. i) Fuzzy set theory is a consistent body of mathematical tools. ii) Although fuzzy sets and probability measures are distinct, several bridges relating them have been proposed that should reconcile opposite points of view ; especially possibility theory stands at the cross-roads between fuzzy sets and probability theory. iii) Mathematical objects that behave like fuzzy sets exist in probability theory. It does not mean that fuzziness is reducible to randomness. Indeed iv) there are ways of approaching fuzzy sets and possibility theory that owe nothing to probability theory. Interpretations of probability theory are multiple especially frequentist versus subjectivist views (Fine [31]) ; several interpretations of fuzzy sets also exist. Some interpretations of fuzzy sets are in agreement with probability calculus and some are not. The paper is structured as follows : first we address some classical misunderstandings between fuzzy sets and probabilities. They must be solved before any discussion can take place. Then we consider probabilistic interpretations of membership functions, that may help in membership function assessment. We also point out nonprobabilistic interpretations of fuzzy sets. The next section examines the literature on possibility-probability transformati...

this paper that the probabilistic representations and the possibilistic ones are not just two equivalent representations of uncertainty. Hence there should be no symmetry between the two mutual conversion procedures. The possibilistic representation is weaker because it explicitly handles imprecision (e.g. incomplete knowledge) and because possibility measures are based on an ordering structure rather than an additive one. Turning a probability measure into a possibility measure may be useful in the presence of other weak sources of information, or when computing with possibilities is simpler than computing with probabilities. Turning a possibility measure into a probability measure might be of interest in the scope of decision-making (Smets, 1990). The next section suggests that the transformations should be guided by two different information principles : the principle of insufficient reason from possibility to probability, and the principle of maximum specificity from probability to possibility. The first principle aims at finding a probability measure which preserves the uncertainty of choice between outcomes, while the second principle aims at finding the most informative possibility distribution, under the constraints dictated by the possibility/probability consistency principle. The paper then proposes two transformations that obey these principles. In the discrete case they are already known. But here, results in the continuous case are given. It is pointed out that these transformations are not related to each other, and the converse transformations are shown to be inadequate. In the last section we discuss the relationship between our approach and other works pertaining to the same topic. Some lines of research are considered.

A quantified model to represent uncertainty is incomplete if its use in a decision environment is not explained. When belief functions were first introduced to represent quantified uncertainty, no associated decision model was proposed. Since then, it became clear that the belief functions meaning is multiple. The models based on belief functions could be understood as an upper and lower probabilities model, as the hint model, as the transferable belief model and as a probability model extended to modal propositions. These models are mathematically identical at the static level, their behaviors diverge at their dynamic level (under conditioning and/or revision). For decision making, some authors defend that decisions must be based on expected utilities, in which case a probability function must be determined. When uncertainty is represented by belief functions, the choice of the appropriate probability function must be explained and justified. This probability function doe...

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. 1.

This paper advocates the use of non-purely probabilistic approaches to higher-order uncertainty. One of the major arguments of Bayesian probability proponents is that representing uncertainty is always decision-driven and as a consequence, uncertainty should be represented by probability. Here we argue that representing partial ignorance is not always decision-driven. Other reasoning tasks such as belief revision for instance are more naturally carried out at the purely cognitive level. Conceiving knowledge representation and decision-making as separate concerns opens the way to non-purely probabilistic representations of incomplete knowledge. It is pointed out that within a numerical framework, two numbers are needed to account for partial ignorance about events, because on top of truth and falsity, the state of total ignorance must be encoded independently of the number of underlying alternatives. The paper also points out that it is consistent to accept a Bayesian view of decision-m...

This paper explains how multisensor data fusion and target identification can be performed within the transferable belief model, a model for the representation of quantified uncertainty based on belief functions. The paper is presented in two parts: methodology and application. In this part, we present the underlying theory, in particular the General Bayesian Theorem needed to transform likelihoods into beliefs and the pignistic transformation needed to build the probability measure required for decision making. We end with a simple example. More sophisticated examples and some comparative studies are presented in Part II. The results presented here can be extended directly to many problems of data fusion and diagnosis.

New semantics for numerical values given to possibility measures are provided. For epistemic possibilities, the new approach is based on the semantics of the transferable belief model, itself based on betting odds interpreted in a less drastic way than what subjective probabilities presupposes. It is shown that the least informative among the belief structures that are compatible with prescribed betting rates is nested, i.e. corresponds to a possibility measure. It is also proved that the idempotent conjunctive combination of two possibility measures corresponds to the hypercautious conjunctive combination of the belief functions induced by the possibility measures. This view di#ers from the subjective semantics first proposed by Giles and relying on upper and lower probability induced by non-exchangeable bets. For objective possibility degrees, the semantics is based on the most informative possibilistic approximation of a probability measure derived from a histogram. The motivation for this semantics is its capability to extend a wellknown kind of confidence intervals around the mode of a distribution to a fuzzy confidence interval. We show how the idempotent disjunctive combination of possibility functions is related to the convex mixture of probability distributions.

The mathematical theory of evidence has been introduced by Glenn Shafer in 1976 as a new approach to the representation of uncertainty. This theory can be represented under several distinct but more or less equivalent forms. Probabilistic interpretations of evidence theory have their roots in Arthur Dempster's multivalued mappings of probability spaces. This leads to random set and more generally to random lter models of evidence. In this probabilistic view evidence is seen as more or less probable arguments for certain hypotheses and they can be used to support those hypotheses to certain degrees. These degrees of support are in fact the reliabilities with which the hypotheses can be derived from the evidence. Alternatively, the mathematical theory of evidence can be founded axiomatically on the notion of belief functions or on the allocation of belief masses to subsets of a frame of discernment. These approaches aim to present evidence theory as an extension of probability theory. Evidence theory has been used to represent uncertainty in expert systems, especially in the domain of diagnostics. It can be applied to decision analysis and it gives a new perspective for statistical analysis. Among its further applications are image processing, project planing and scheduling and risk analysis. The computational problems of evidence theory