A Philosophical Foundation of Non-Additive Measure and Probability
- 97 Downloads
In this paper, non-additivity of a set function is interpreted as a method to express relations between sets which are not modeled in a set theoretic way. Drawing upon a concept called “quasi-analysis” of the philosopher Rudolf Carnap, we introduce a transform for sets, functions, and set functions to formalize this idea. Any image-set under this transform can be interpreted as a class of (quasi-)components or (quasi-)properties representing the original set. We show that non-additive set functions can be represented as signed σ-additive measures defined on sets of quasi-components. We then use this interpretation to justify the use of non-additive set functions in various applications like for instance multi criteria decision making and cooperative game theory. Additionally, we show exemplarily by means of independence, conditioning, and products how concepts from classical measure and probability theory can be transfered to the non-additive theory via the transform.
Keywordsconditioning independence Möbius transform non-additive measure products quasi-analysis
Unable to display preview. Download preview PDF.
- Brüning, M. (2003). Products of Monotone Measures, Möbius Transform and k-monotonicity. Dissertation thesis, Universität Bremen.Google Scholar
- Carnap, R. 1928Der logische Aufbau der WeltMeinerHamburgGoogle Scholar
- Choquet, G. 1953–1954Theory of CapacitiesAnnales de l’Institut Fourier5131295Google Scholar
- Denneberg, D. (1994, 1997 2nd. ed.), Non-additive Measure and Integral Kluwer, Dordrecht.Google Scholar
- Gilboa, I., Schmeidler, D. 1995Canonical representation of set functionsMathematics of Operations Research20197212Google Scholar
- Shafer, G. 1976A Mathematical Theory of EvidencePrinceton Univ. PressPrinceton, NJGoogle Scholar
- Walley, P., Fine, T.L. 1982Towards a frequentist theory of upper and lower probabilityAnnals of Statistics10741761Google Scholar