A New Method to Determine Evidence Distance

Shi, Chao; Chen, Yongmei; Lu, Yao

doi:10.1109/cise.2010.5676947

Cited by 8 publications

(7 citation statements)

References 6 publications

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“…A number of norms for belief functions have been introduced [71,90,93,126], as a tool for assessing the level of conflict between different bodies of evidence, for approximating a b.f. using a different uncertainty measure and so on.…”

Section: Distances In Evidence Theorymentioning

confidence: 99%

A geometric approach to conditioning belief functions

Cuzzolin

2021

Preprint

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Conditioning is crucial in applied science when inference involving time series is involved. Belief calculus is an effective way of handling such inference in the presence of epistemic uncertainty -unfortunately, different approaches to conditioning in the belief function framework have been proposed in the past, leaving the matter somewhat unsettled. Inspired by the geometric approach to uncertainty, in this paper we propose an approach to the conditioning of belief functions based on geometrically projecting them onto the simplex associated with the conditioning event in the space of all belief functions. We show here that such a geometric approach to conditioning often produces simple results with straightforward interpretations in terms of degrees of belief. This raises the question of whether classical approaches, such as for instance Dempster's conditioning, can also be reduced to some form of distance minimisation in a suitable space. The study of families of combination rules generated by (geometric) conditioning rules appears to be the natural prosecution of the presented research.

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Section: Distances In Evidence Theorymentioning

confidence: 99%

A geometric approach to conditioning belief functions

Cuzzolin

2021

Preprint

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“…As to what distances are the most appropriate, Jousselme et al [29] have recently conducted a nice survey of the distance or similarity measures so far introduced in belief calculus, come out with an interesting classification, and proposed a number of generalizations of known measures. Other similarity measures between belief functions have been proposed by Shi et al [30], Jiang et al [31], and others [32], [33]. Many of these measures can be in principle employed to define conditional belief functions, or to approximate belief functions by necessity or probability measures.…”

Section: Introductionmentioning

confidence: 99%

“…For instance, generalizations to belief functions of the classical Kullback-Leibler divergence between probability distributions or other measures based on information theory such as fidelity and entropy-based norms [41] can be studied. Many other similarity measures have indeed been proposed [30], [31], [32], [33]. The application to the approximation problem of similarity measures more specific to belief functions or inspired by classical probability is a huge task, of which this paper is just a first step.…”

Section: Introductionmentioning

confidence: 99%

$L_{p}$ Consonant Approximations of Belief Functions

Cuzzolin

2014

IEEE Trans. Fuzzy Syst.

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In this paper we solve the problem of approximating a belief measure with a necessity measure or "consonant belief function" in a geometric framework. Consonant belief functions form a simplicial complex in both the space of all belief functions and the space of all mass vectors: partial approximations are first sought in each component of the complex, while global solutions are selected among them. As a first step in this line of study, we seek here approximations which minimize L p norms. Approximations in the mass space can be interpreted in terms of mass redistribution, while approximations in the belief space generalize the maximal outer consonant approximation. We compare them with each other and with other classical approximations, and illustrate them with the help of a running example. Index TermsTheory of evidence, possibility theory, consonant belief functions, geometric approach, simplicial complex, (outer) consonant approximation, isopignistic function, L p norms.

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“…Many of these measures could be in principle employed to define indifferently consistent transformations or conditional belief functions [23], or approximate belief functions by necessity or probability measures. Other similarity measures between belief functions have been proposed by Shi et al [24], Jiang et al [25], and others [26,27].…”

Section: Introductionmentioning

confidence: 99%

On Consistent Approximations of Belief Functions in the Mass Space

Cuzzolin

2011

Lecture Notes in Computer Science

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Consistent belief functions represent collections of coherent or non-contradictory pieces of evidence, but most of all they are the counterparts of consistent knowledge bases in belief calculus. The use of consistent transformations cs[·] in a reasoning process to guarantee coherence can therefore be desirable, and generalizes similar techniques in classical logics. Transformations can be obtained by minimizing an appropriate distance measure between the original belief function and the collection of consistent ones. We focus here on the case in which distances are measured using classical L p norms, in both the "mass space" and the "belief space" representation of belief functions. While mass consistent approximations reassign the mass not focussed on a chosen element of the frame either to the whole frame or to all supersets of the element on an equal basis, approximations in the belief space do distinguish these focal elements according to the "focussed consistent transformation" principle. The different approximations are interpreted and compared, with the help of examples.

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A New Method to Determine Evidence Distance

Cited by 8 publications

References 6 publications

A geometric approach to conditioning belief functions

A geometric approach to conditioning belief functions

$L_{p}$ Consonant Approximations of Belief Functions

On Consistent Approximations of Belief Functions in the Mass Space

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