Coherent Conditional Plausibility: A Tool for Handling Fuzziness and Uncertainty Under Partial Information

Abstract

Non-additive measures, such as plausibility, are meaningful when only a partial or indirect information on the events of interest is available, or when imprecision and ambiguity of agents are considered. Our main aim is to study inferential processes, like the Bayesian one, when the information is expressed in natural language and the uncertainty measure is either partially or imprecisely assessed. We deal with partial assessments consistent with a conditional plausibility, and adopt the interpretation of the membership of a fuzzy set in terms of coherent conditional plausibility, regarded as a function of the conditioning events. This kind of interpretation, inspired to that given in terms of coherent conditional probability, is particularly useful for computing the measure of the uncertainty of fuzzy events, when the knowledge on the variable is imprecise and can be managed with a non-additive measure of uncertainty. A simple situation related to a Zadeh’s example can be the following: a ball will be drawn from an urn containing balls of different colours and different diameters, but one knows only the distribution of the different colours. The aim is to compute the uncertainty measure of the fuzzy event “a small ball is drawn”, taking in considerations possible logical constrains among the colours and the diameters.