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

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, Lama...

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Vydané v:	International journal of approximate reasoning Ročník 92; s. 49 - 65
Hlavní autori:	Jiroušek, Radim, Shenoy, Prakash P.
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Elsevier Inc 01.01.2018
Predmet:	Dempster's rule of combination Dempster–Shafer theory of belief functions Dempster–Shafer theory semantics Maximum entropy property Plausibility transform of a belief function Dempster–Shafer theory of belief functions Dempster's rule of combination Plausibility transform of a belief function Maximum entropy property Dempster–Shafer theory semantics
ISSN:	0888-613X, 1873-4731
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Popis
Shrnutí:	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. •We state six desired properties that should be satisfied by an entropy function in the Dempster–Shafer theory.•Three of the six desired properties are different from the five properties defined by Klir and Wierman.•We show that all previously defined entropy functions do not meet all six desired properties.•We prove that our new definition of entropy satisfies all six desired properties.•We discuss also the subadditivity property and show that our definition does not satisfy it.
ISSN:	0888-613X 1873-4731
DOI:	10.1016/j.ijar.2017.10.010