Entailment Principle for Measure-Based Uncertainty

We describe the use of monotonic set measures, which are sometimes referred to as fuzzy measures, in the representation of uncertain information about a variable. The representation of significant types of uncertain information, such as probabilistic, possibilistic, and others, is discussed. The concept of the dual of a measure is introduced and is used to define the measures of opportunity and assurance associated with the occurrence of an event. We observe that these generalize the ideas of possibility and necessity in possibility theory, as well as the ideas of plausibility and belief in Dempster-Shafer theory. Using the measures of opportunity and assurance, we are able to introduce a concept of entailment between two uncertainty representations, which generalizes Zadeh´s idea of entailment between possibility distributions. We show, using the idea of entailment, how to relate possibilistic and probabilistic uncertainty and discuss a potential application in hard/soft information fusion. We, finally, consider the measure representation of joint variables.