Abstract-Random variability and imprecision are two distinct facets of the uncertainty affecting parameters that influence the assessment of risk. While random variability can be represented by probability distribution functions, imprecision (or partial ignorance) is better accounted for by possibility distributions (or families of probability distributions). Because practical situations of risk computation often involve both types of uncertainty, methods are needed to combine these two modes of uncertainty representation in the propagation step. A hybrid method is presented here, which jointly propagates probabilistic and possibilistic uncertainty. It produces results in the form of a random fuzzy interval. This paper focuses on how to properly summarize this kind of information; and how to address questions pertaining to the potential violation of some tolerance threshold. While exploitation procedures proposed previously entertain a confusion between variability and imprecision, thus yielding overly conservative results, a new approach is proposed, based on the theory of evidence, and is illustrated using synthetic examples.
This article deals with the compact representation of incomplete probabilistic knowledge which can be encountered in risk evaluation problems, for instance in environmental studies. Various kinds of knowledge are considered such as expert opinions about characteristics of distributions or poor statistical information. Our approach is based on probability families encoded by possibility distributions and belief functions. In each case, a technique for representing the available imprecise probabilistic information faithfully is proposed, using different uncertainty frameworks (possibility theory, probability theory, belief functions...). Moreover the use of probability-possibility transformations enables confidence intervals to be encompassed by cuts of possibility distributions, thus making the representation stronger. The respective appropriateness of pairs of cumulative distributions, continuous possibility distributions or discrete random sets for representing information about the mean value, the mode, the median and other fractiles of ill-known probability distributions is discussed in detail.
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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