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

TL;DR

This paper develops a comprehensive framework combining belief functions, fuzzy sets, and evidence theory to better represent and integrate uncertain and fuzzy evidence, with applications to statistical inference.

Contribution

It introduces a general theory of epistemic random fuzzy sets that unifies Dempster-Shafer theory and possibility theory, enabling advanced evidence combination.

Findings

Unified framework for uncertain and fuzzy evidence

Reconciliation of possibilistic likelihood with Bayesian inference

Application to statistical inference demonstrating practical utility

Abstract

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