Probability-Possibility Transformation under Perspective of Random-fuzzy Dual Interpretation of Unknown Uncertainty

Abstract

Information-preservation is recognized as the only principle for probability-possibility transformation in this work and the normalized transformation is the right method. This is based on the viewpoint that the reason we can transfer probability and possibility is that we believe the uncertainty being handled can be information-equivalently described by both probability and possibility. That viewpoint is endorsed by the random-fuzzy dual interpretation of unknown uncertainty, which says that unknown uncertainty being estimated could be interpreted as either randomness or fuzziness, depending on the available prior information and the perspective of cognition and modeling. Information of uncertain variable is defined in this work as its distribution. The suggested information-preservation principle is different from Klir’s principle, which is in fact an entropy-preservation principle. Then we investigated the problem of information preservation and propagation in parallel probability-possibility systems. By parallel, we mean the two uncertainty systems have the same priori information. After uncertainty propagation the two parallel systems will generally bifurcate, which means information preservation only holds locally between the two parallel systems. This observation accords with our intuition since probability and possibility use different normalizations as well as different disjunctive operators, which makes them two different uncertainty systems appropriate for randomness and fuzziness, respectively.