Remarks on “Measuring Ambiguity in the Evidence Theory”

Klir, George J.; Lewis, Howard B.

doi:10.1109/tsmca.2008.923066

Cited by 55 publications

(24 citation statements)

References 6 publications

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“…Subadditivity of the pignistic entropy for mass functions in multi-variate systems was claimed in [20], but a flaw in the argument was later pointed out by [21]. The latter reference even presented a counter-example for this property, which we here translate from mass functions to hints.…”

Section: Subadditivity Of Hints and Mass Functionsmentioning

confidence: 74%

“…Computing its marginal entropy with respect to the frame of discernment uncovers an uncertainty measure for the DempsterShafer theory called pignistic entropy, which was proposed in the literature as a counter-example for the uniqueness of the aggregate uncertainty. Although this functional gets over the known limitations of the aggregate uncertainty, it was later shown to violate the fundamental property of subadditivity and was therefore rejected, see [21]. Abstracting from individual hints to equivalence classes of hints and therefore to mass functions suppresses probabilistic information, which gives a new explanation for this lack of subadditivity.…”

Section: Discussionmentioning

confidence: 99%

“…Given that the range of the hints entropy is different, this question remains open. However, Jousselme [20] originally claimed that the pignistic entropy disproves uniqueness of the aggregate uncertainty, but this paper contained a flawed proof of subadditivity, which was later pointed out by [21]. See Example 8 for a counter-example.…”

Section: The Aggregate Uncertaintymentioning

confidence: 99%

“…As stated above, on this larger space the hints entropy satisfies all classical requirements for an uncertainty measure. But it will also be shown that the restriction to the space of mass functions reveals a different measure called pignistic entropy, which was proposed as an uncertainty measure in the literature [20] but later rejected as it violates subadditivity [21]. Thanks to the viewpoint of hints we can explain the lack of subadditivity by the loss of probabilistic information in the step from individual hints to equivalence classes of hints.…”

Section: Introductionmentioning

confidence: 97%

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Generalized information theory for hints

Pouly

Kohlas

Ryan

2013

International Journal of Approximate Reasoning

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This paper develops a new uncertainty measure for the theory of hints that complies with the established semantics of statistical information theory and further satisfies all classical requirements for such a measure imposed in the literature. The proposed functional decomposes into conversant uncertainty measures and therefore discloses a new interpretation of the latters as well. By abstracting to equivalence classes of hints we transport the new measure to mass functions in Dempster-Shafer theory and analyse its relationship with the aggregate uncertainty, which currently is the only known functional for the DempsterShafer theory of evidence that satisfies the same set of properties. Moreover, the perspective of hints reveals that the standard independence notion in Dempster-Shafer theory called non-interactivity corresponds to an amalgamation of probabilistic independence and qualitative independence between frames of discernment. All results in this paper are developed for arbitrary families of compatible frames generalizing the very specialized multi-variate systems that are usually studied in information theory.

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Section: Subadditivity Of Hints and Mass Functionsmentioning

confidence: 74%

Section: Discussionmentioning

confidence: 99%

Section: The Aggregate Uncertaintymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

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Generalized information theory for hints

Pouly

Kohlas

Ryan

2013

International Journal of Approximate Reasoning

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“…Although Jousselme et al claim that H j (m) satisfies the subadditivity property (Eq. (23)), a counter-example is provided in [23]. One basic assumption behind H j (m) is that BetP m is an appropriate probabilistic representation of the uncertainty in m in the DS theory.…”

Section: Previous Definitions Of Entropy Of Bpas In the Ds Theorymentioning

confidence: 99%

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

Jiroušek

Shenoy

2018

International Journal of Approximate Reasoning

132

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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

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