Reasoning with fuzzy and uncertain evidence using epistemic random fuzzy sets: general framework and practical models

TL;DR

This work develops a general theory of epistemic random fuzzy sets that unifies DS belief functions and possibility theory, enabling reasoning with both uncertain and fuzzy evidence. It introduces a generalized product-intersection rule for combining independent epistemic random fuzzy sets and develops practical models—Gaussian random fuzzy numbers and Gaussian random fuzzy vectors—for closed-form combination, projection, and vacuous extension. The framework covers statistical inference by integrating likelihood-based belief and plausibility measures and clarifies how to fuse crisp and fuzzy information without the inconsistencies of traditional Dempster-Shafer updates. Overall, the approach provides a versatile, mathematically tractable toolkit for representing and manipulating mixed probabilistic and fuzzy uncertainty in diverse applications, including inference and prediction under uncertainty.

Abstract

We introduce a general theory of epistemic random fuzzy sets for reasoning with fuzzy or crisp evidence. This framework generalizes both the Dempster-Shafer theory of belief functions, and possibility theory. Independent epistemic random fuzzy sets are combined by the generalized product-intersection rule, which extends both Dempster's rule for combining belief functions, and the product conjunctive combination of possibility distributions. We introduce Gaussian random fuzzy numbers and their multi-dimensional extensions, Gaussian random fuzzy vectors, as practical models for quantifying uncertainty about scalar or vector quantities. Closed-form expressions for the combination, projection and vacuous extension of Gaussian random fuzzy numbers and vectors are derived.

Figures (2)

• Figure 1: Lower and upper cdf's for the random fuzzy numbers studied in Examples \ref{['ex:RFS']} and \ref{['ex:RFS_BelPl']}, with $\mu=0$, $\sigma=1$, and $a=0.5$ (blue curves) or $a=1.5$ (red curves). The Gaussian cdf corresponding to $a=0$ is shown as a broken line.
• Figure 2: (a): Two Gaussian possibility distributions (black solid curves) with their normalized product intersection (red broken curve) and the contour function of the combined random set (blue solid curve). (b): Lower and upper cdf's of the combined possibility distribution (red broken curves) and of the combined random set (blue solid curves).

Theorems & Definitions (50)

• Proposition 1
• proof
• Example 1
• Proposition 2
• proof
• Example 2
• Example 3
• Proposition 3
• Example 4
• Example 5