Required mathematical properties and behaviors of uncertainty measures on belief intervals

Abstract

The Dempster–Shafer theory of evidence (DST) has been widely used to handle uncertainty‐based information. It is based on the concept of basic probability assignment (BPA). Belief intervals are easier to manage than a BPA to represent uncertainty‐based information. For this reason, several uncertainty measures for DST recently proposed are based on belief intervals. In this study, we carry out a study about the crucial mathematical properties and behavioral requirements that must be verified by every uncertainty measure on belief intervals. We base on the study previously carried out for uncertainty measures on BPAs. Furthermore, we analyze which of these properties are satisfied by each one of the uncertainty measures on belief intervals proposed so far. Such a comparative analysis shows that, among these measures, the maximum of entropy on the belief intervals is the most suitable one to be employed in practical applications since it is the only one that satisfies all the required mathematical properties and behaviors.

An Effective Measure of Uncertainty of Basic Belief Assignments

“…Remark 1: It is worth noting that we do not have specified a priori what should be the range of an effective MoU in contrary to some axiomatic attempts made by different authors as reported, for instance, in [12]. We consider that the choice of the range must not be chosen a priori.…”

“…To justify our choice, just consider a simple example with |Θ| v only related to the 8 elements of Θ ′ ? We do not see any solid theoretical reason, nor intuitive reason, for justifying and requiring the subadditivity desideratum in the general framework of belief functions, and to select it as an axiom to satisfy in general as done in [12]. Unlike Vejnarova and Klir opinions [15] (p.28) (and some authors following them), we do not consider that the meaningful (or effective) measure of uncertainty of basic belief assignment must satisfy the sub-additivity desideratum in general.…”

A new correlation belief function in Dempster-Shafer evidence theory and its application in classification

“…Deng and Wang 66 measure the Hellinger distance between the belief interval and the most uncertain interval for each single case as the total uncertainty. Besides, for the existing methods of uncertainty measurement, Moral-García and Abellán 67 pointed out that the maximum value of entropy on the belief interval is the most suitable way of measurement for practical applications because of its excellent mathematical properties. This work focuses on information fusion in classification problem with respect to uncertainty management with a new correlation factor in the framework of D-S evidence theory.…”

A New Correlation Measure for Belief Functions and Their Application in Data Fusion

“…The uncertainty of information is mainly manifested as vagueness, unknown, inaccuracy, etc. [ 1 , 2 ]. At present, related methods of uncertain information processing have been widely applied to decision making [ 3 , 4 ], image classification [ 5 , 6 ], and many other fields [ 7 , 8 , 9 ].…”