A MATHEMATICAL ANALYSIS OF INFORMATION-PRESERVING TRANSFORMATIONS BETWEEN PROBABILISTIC AND POSSIBILISTIC FORMULATIONS OF UNCERTAINTY

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Abstract
It is now generally recognized that uncertainty can be formalized in different mathematical theories. Two of these theories, on which we focus in this paper, are probability theory and possibility theory. The paper deals with transformations from probabilistic formalizations of uncertainty into their possibilistic counterparts that contain the same amount of uncertainty and, consequently, the same amount of information (expressed as a reduction of uncertainty) as well; it also deals with the inverse uncertainty and information preserving transformations. Since well-justified and unique measures of uncertainty (and information) are now well established in both probability theory and possibility theory, the transformations are well defined. Mathematical properties of the transformations are analyzed in the paper under the assumption that probabilities and possibilities are connected via interval or log-interval scales. The primary results are: (i) the interval scale transformation that preserves information exists and is unique only from probability theory to possibility theory, but the inverse transformation does not always exist; (ii) the log-interval scale transformation exists and is unique in both directions; and (iii) the log-interval scale transformation satisfies the probability-possibility consistency requirement.

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