## Properties of measures of information in evidence and possibility theories.(English)Zbl 0633.94009

In this paper it is investigated the additivity and monotonicity properties of different measures of information which have been recently introduced in the framework of Shafer’s evidence theory. The existence of these properties shows that it is possible to extend information theory in a nice way beyond its probabilistic setting.
Reviewer: L.Pardo

### MSC:

 94A17 Measures of information, entropy 94D05 Fuzzy sets and logic (in connection with information, communication, or circuits theory)
Full Text:

### References:

 [1] Aczél, L., Lectures on functional equations and applications, (1972), Academic Press New York [2] Arora, P.N., On characterizing some generalizations of Shannon’s entropy, Inform. sci., 21, 13-22, (1980) · Zbl 0469.94007 [3] () [4] De Luca, A.; Termini, S., A definition of a non-probabilistic entropy in the setting of fuzzy set theory, Inform. and control, 20, 301-312, (1972) · Zbl 0239.94028 [5] Dempster, A.P., Upper and lower probabilities induced by a multi-valued mapping, Ann. math. statist., 38, 325-339, (1967) · Zbl 0168.17501 [6] Di Nola, A.; Sessa, S., On the fuzziness measure and negation in totally ordered lattices, Busefal, 8, 68-77, (1981) [7] Dubois, D.; Prade, H., Fuzzy sets and systems: theory and applications, (1980), Academic Press New York · Zbl 0444.94049 [8] Dubois, D.; Prade, H., On several representations of an uncertain body of evidence, (), 167-181 [9] Dubois, D.; Prade, H.; Dubois, D.; Prade, H., A note on measures of specificity for fuzzy sets, Internat. J. gen. systems, Preprint in BUSEFAL, 19, 83-89, (1984) · Zbl 0555.94028 [10] Dubois, D.; Prade, H., Théorie des possibilités, () [11] Dubois, D.; Prade, H., Mesures d’information pour LES distributions de possibilité et LES fonctions de croyance, (), 67-80, GR 22, CNRS [12] Dubois, D.; Prade, H., Additivity and monotonicity of measures of information defined in the setting of Shafer’s evidence theory, Busefal, 24, 64-76, (1985) · Zbl 0607.94002 [13] Dubois, D.; Prade, H., A set-theoretic view of belief functions: logical operations and approximations by fuzzy sets, Internat. J. gen. systems, 12, 193-226, (1986) [14] Higashi, M.; Klir, G.J., On measures of fuzziness and fuzzy complements, Internat. J. gen. systems, 8, 115-124, (1982) · Zbl 0484.94047 [15] Higashi, M.; Klir, G.J., Measures of uncertainty and information based on possibility distributions, Internat. J. gen. systems, 9, 43-58, (1983) · Zbl 0497.94008 [16] Höhle, U., Fuzzy plausibility measures, (), 7-30 [17] Höhle, U., Entropy with respect to plausibility measures, (), 167-169 [18] Höhle, U., Fuzzy filters: A generalization of credibility measures, (), 111-114 [19] Jaynes, E.T., Where do we stand on maximum entropy, () · Zbl 0687.60085 [20] Kaufmann, A., Introduction à la théorie des sous-ensembles, () · Zbl 0302.02023 [21] Klir, G., Where do we stand on measures of uncertainty, ambiguity, fuzziness, and the like?, Fuzzy sets and systems, 24, 141-160, (1987), (this issue) · Zbl 0633.94026 [22] Nguyen, H.T., On entropy of random sets and possibility distributions, () · Zbl 0651.60002 [23] Oblow, E.M., A hybrid uncertainty theory, (), 1193-1201 [24] Prade, H., Reasoning with fuzzy default values, (), 191-197 [25] Prade, H.; Testemale, C., Representation of soft constraints and fuzzy attribute values by means of possibility distributions in data bases, () [26] Shafer, G., A mathematical theory of evidence, (1976), Princeton University Press · Zbl 0359.62002 [27] Shafer, G., Belief functions and possibility measures, () · Zbl 0655.94025 [28] Smets, P., Information content of an evidence, Internat. J. man-machine stud., 19, 33-43, (1983) [29] Trillas, E., An approach to fuzziness in the setting of łukasiewicz logic, (), 222-226 [30] Trillas, E.; Riera, T., Entropies of finite fuzzy sets, Inform. sci., 15, 2, 159-168, (1978) · Zbl 0436.94012 [31] Yager, R.R.; Yager, R.R., On the measure of finite fuzziness and negation, part 2, lattices, Internat. J. gen. systems, Inform. and control, 44, 236-260, (1980) · Zbl 0429.04008 [32] Yager, R.R., Measurement of properties on fuzzy sets and possibility distribution, (), 211-222 [33] Yager, R.R., Measures of fuzziness based on t-norms, Stochastica, 6, 3, 207-229, (1982) · Zbl 0541.94043 [34] Yager, R.R., Measuring tranquality and anxiety in decision-making: an application of fuzzy, Internat. J. gen. systems, 8, 139-146, (1982) · Zbl 0487.90007 [35] Yager, R.R., Entropy and specificity in a mathematical theory of evidence, Internat. J. gen. systems, 9, 249-260, (1983) · Zbl 0521.94008 [36] Yager, R.R., On different classes of linguistic variables defined via fuzzy subsets, Kybernetes, 13, 103-110, (1984) · Zbl 0544.03008 [37] Yager, R.R., The entailment principle for Dempster-Shafer granules, () · Zbl 0643.94054 [38] Zadeh, L.A., Fuzzy sets, Inform. and control, 8, 338-353, (1965) · Zbl 0139.24606 [39] Zadeh, L.A., Fuzzy sets as a basis for a theory of possibility, Fuzzy sets and systems, 1, 1, 3-28, (1978) · Zbl 0377.04002 [40] Zadeh, L.A., Fuzzy sets and information granularity, (), 3-18 · Zbl 0377.04002 [41] Zadeh, L.A., A simple view of the Dempster-Shafer theory of evidence, () [42] Nguyen, H.T., On random sets and belief functions, J. math. anal. appl., 65, 531-542, (1978) · Zbl 0409.60016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.