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We revisit Zadeh's notion of "evidence of the second kind" and show that itprovides the foundation for a general theory of epistemic random fuzzy sets,which generalizes both the Dempster-Shafer theory of belief functions andpossibility theory. In this perspective, Dempster-Shafer theory deals withbelief functions generated by random sets, while possibility theory deals withbelief functions induced by fuzzy sets. The more general theory allows us torepresent and combine evidence that is both uncertain and fuzzy. We demonstratethe application of this formalism to statistical inference, and show that itmakes it possible to reconcile the possibilistic interpretation of likelihoodwith Bayesian inference.

Belief functions induced by random fuzzy sets: A general framework for representing uncertain and fuzzy evidence