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OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. In information processing tasks, sets may have a conjunctive or a disjunctive reading. In the conjunctive reading, a set represents an object of interest and its elements are subparts of the object, forming a composite description. In the disjunctive reading, a set contains mutually exclusive elements and refers to the representation of incomplete knowledge. It does not model an actual object or quantity, but partial information about an underlying object or a precise quantity. This distinction between what we call ontic vs. epistemic sets remains valid for fuzzy sets, whose membership functions, in the disjunctive reading are possibility distributions, over deterministic or random values. This paper examines the impact of this distinction in statistics. We show its importance because there is a risk of misusing basic notions and tools, such as conditioning, distance between sets, variance, regression, etc. when data are set-valued. We discuss several examples where the ontic and epistemic points of view yield different approaches to these concepts.

This paper discusses some models of Imprecise Probability Theory obtained by propagating uncertainty in risk analysis when some input parameters are stochastic and perfectly observable, while others are either random or deterministic, but the information about them is partial and is represented by possibility distributions. Our knowledge about the probability of events pertaining to the output of some function of interest from the risk analysis model can be either represented by a fuzzy probability or by a probability interval. It is shown that this interval is the average cut of the fuzzy probability of the event, thus legitimating the propagation method. Besides several independence assumptions underlying the joint probability-possibility propagation methods are discussed and illustrated by a motivating example.

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In this paper we propose a way of measuring the difference between two fuzzy sets by means of a function which we will call divergence. We define this concept by means of a group of natural axioms and we study in detail the most important classes of such measures, those which have the local property.

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