A new definition of entropy of belief functions in the Dempster–Shafer theory

Abstract

We propose a new definition of entropy of basic probability assignments (BPAs) in the Dempster-Shafer (DS) theory of belief functions, which is interpreted as a measure of total uncertainty in the BPA. Our definition is different from those proposed by Höhle, Smets, Yager, Nguyen, Dubois-Prade, Lamata-Moral, Klir-Ramer, Klir-Parviz, Pal et al., Maeda-Ichihashi, Harmanec-Klir, Abellán-Moral, Jousselme et al., Pouly et al., and Deng. We state a list of six desired properties of entropy for DS belief functions theory, four of which are motivated by Shannon's definition of entropy of probability functions, and the remaining two are requirements that adapt this measure to the philosophy of the DS theory. Three of our six desired properties are different from the five properties proposed by Klir and Wierman. We demonstrate that our definition satisfies all six properties in our list, whereas none of the existing definitions do. Our new definition has two components. The first component is Shannon's entropy of an equivalent probability mass function obtained using the plausibility transform, which constitutes the conflict measure of entropy. The second component is Dubois-Prade's definition of entropy of basic probability assignments in the DS theory, which constitutes the non-specificity measure of entropy. Our new definition is the sum of these two components. Our definition does not satisfy the subadditivity property. Whether there exists a definition that satisfies our six properties plus subadditivity remains an open question. 1

A total uncertainty measure for D numbers based on belief intervals

“…Once the nonexclusive degrees with respect to every feature are determined, we can calculate the belief intervals for each D number according to Equations (13) and (15), and then the uncertainty degree of associated D number according to Equations (16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27) to (19). Finally, the average uncertainty degree of these D numbers associated with a same feature can be derived, as given in Table 2.…”

Evidence combination using OWA‐based soft likelihood functions

“…Belief entropy is defined to measure the uncertainty degree of a BPA, and various entropy functions have been developed for DST . Recently, a new definition of entropy has been proposed by Jiroušek and Shenoy with some good properties, which was defined as follows:…”

“…Also, an appropriate dissimilarity measure could be selected according to the actual situation. In this study, the entropy functions defined by Jiroušek and Shenoy and the Jousselme evidence distance are considered as the uncertainty and distance measures of sources of evidence due to their good properties.…”

Dependent evidence combination based on decision‐making trial and evaluation laboratory method

“…Many math tools are presented to model uncertainty, such as Z numbers, D numbers, and entropy model . Among then, evidence theory is widely used due to its efficiency to deal with uncertainty.…”

“…Many math tools are presented to model uncertainty, such as Z numbers, [53][54][55] D numbers, [56][57][58][59][60][61] and entropy model. [62][63][64][65][66][67] Among then, evidence theory is widely used due to its efficiency to deal with uncertainty. In this section, the traditional D-S evidence, 1 the Pearson correlation coefficient, 52 the Spearman correlation coefficient, 68 and the DEMATEL method are introduced.…”