Uncertainty measures in a generalized theory of evidence

Abstract

Epistemic Random Fuzzy Set theory is an extension of Dempster-Shafer and possibility theories in which pieces of evidence are represented by random fuzzy sets and combined by the product-intersection rule, an extension of Dempster's rule and the product combination of possibility distributions. We propose a measure of imprecision and a measure of conflict for random fuzzy sets, uniquely characterized by minimal sets of requirements. Both measures have simple expressions involving only the contour function: in the finite case, imprecision is measured by the logarithm of the sum of the plausibilities of the singletons, while conflict is measured by the negative logarithm of the maximum plausibility over the singletons. These definitions can be easily carried over to random fuzzy sets in continuous spaces, allowing us to define the imprecision and conflict of Gaussian random fuzzy numbers and extensions. Total uncertainty is defined as the sum of imprecision and conflict. The corresponding measure, referred to as $T$-entropy, happens to be the min-entropy and the nonspecificity measure of, respectively, the probability distribution and the normalized possibility distribution constructed from the contour function. The application of these uncertainty measures to belief elicitation is discussed and illustrated by some examples.