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.