The Sigma-Max System Induced from Randomness and Fuzziness

Mei, Wei; Li, Ming; Cheng, Yijie; Liu, Limin

doi:10.48550/arxiv.2110.07722

Cited by 2 publications

(7 citation statements)

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“…Since random outcomes are mutual exclusive, probability requires that scores of every possible outcome of a random variable should sum to one, which is the key axiom of "additivity" [13][14][15]. In contrast, possibility is marked by the key axiom of "maxitivity" [1,2,6], which says that the possibility of two disjunctive non-exclusive fuzzy concepts is equal to the maximum of the two constituent possibilities. Possibility theory could be regarded as the foundation for fuzzy sets, since membership function of fuzzy sets could be recognized as likelihood function of possibility [16], and composition of fuzzy relations is equivalent to composition of conditional possibilities [5].…”

Section: Clear Boundary Between Non-exclusive Outcomesmentioning

confidence: 99%

“…Remark: A random variable should be modeled by probability and fuzzy variable by possibility [4,6]. Though events in ⊆ 2 and ⊆ 2 are both not mutually exclusive, the structures of random event space ⊆ 2 and fuzzy event space ⊆ 2 are not the same because their corresponding sample spaces and are defined differently [6].…”

Section: Random/fuzzy Variablesmentioning

confidence: 99%

“…The fuzzy uncertainty arisen here has common in essence with that of the natural fuzzy concept such as Young and Senior, i.e., fuzziness is caused by the overlap of their intensions. Here intension indicates the internal content (abstracted feature) of a concept that constitutes its formal definition [6,43].…”

Section: The Dual Interpretation Of Unknown Uncertaintymentioning

confidence: 99%

“…As an alternative method to fuzzy sets for handling fuzzy uncertainty, possibility theory has in recent decades received extensive attention [1][2][3][4][5][6][7][8][9][10][11][12][44][45][46]. It was lately demonstrated that probability and possibility are customized for the measure of randomness and fuzziness, respectively [6]. As two kinds of widely acknowledged fundamental uncertainties of this world, randomness and fuzziness are recently defined in [6,8] as below:…”

Section: Introductionmentioning

confidence: 99%

“…It was lately demonstrated that probability and possibility are customized for the measure of randomness and fuzziness, respectively [6]. As two kinds of widely acknowledged fundamental uncertainties of this world, randomness and fuzziness are recently defined in [6,8] as below:…”

Section: Introductionmentioning

confidence: 99%

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Probability-Possibility Transformation under Perspective of Random-fuzzy Dual Interpretation of Unknown Uncertainty

Mei¹,

Li²,

Xu³

et al. 2022

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Information-preservation is recognized as the only principle for probability-possibility transformation in this work and the normalized transformation is the right method. This is based on the viewpoint that the reason we can transfer probability and possibility is that we believe the uncertainty being handled can be information-equivalently described by both probability and possibility. That viewpoint is endorsed by the random-fuzzy dual interpretation of unknown uncertainty, which says that unknown uncertainty being estimated could be interpreted as either randomness or fuzziness, depending on the available prior information and the perspective of cognition and modeling. Information of uncertain variable is defined in this work as its distribution. The suggested information-preservation principle is different from Klir’s principle, which is in fact an entropy-preservation principle. Then we investigated the problem of information preservation and propagation in parallel probability-possibility systems. By parallel, we mean the two uncertainty systems have the same priori information. After uncertainty propagation the two parallel systems will generally bifurcate, which means information preservation only holds locally between the two parallel systems. This observation accords with our intuition since probability and possibility use different normalizations as well as different disjunctive operators, which makes them two different uncertainty systems appropriate for randomness and fuzziness, respectively.

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