Abstract
A multivalued mapping from a space X to a space S carries a probability measure defined over subsets of X into a system of upper and lower probabilities over subsets of S. Some basic properties of such systems are explored in Sects. 1 and 2. Other approaches to upper and lower probabilities are possible and some of these are related to the present approach in Sect. 3. A distinctive feature of the present approach is a rule for conditioning, or more generally, a rule for combining sources of information, as discussed in Sects. 4 and 5. Finally, the context in statistical inference from which the present theory arose is sketched briefly in Sect. 6.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Dempster, A. P (1966)). New approaches for reasoning towards posterior distributions based on sample data. Ann. Math. Statist. 37 355–374.
Fishburn, Peter C. (1964)). Decision and Value Theory. Wiley, New York.
Good, I. J. (1962)). The measure of a non-measurable set.Logic, Methodology and Philosophy of Science (edited by Ernest Nagel, Patrick Suppes, and Alfred Tarski). Stanford Univ.Press. 319–329.
Koopman, B. O. (1940a). The axioms and algebra of intuitive probability. Ann. Math. 41 269–292.
Koopman, B. O. (1940b). The bases of probability. Bull. Amer. Math. Soc. 46 763–774.
Smith, C. A. B. (1961)). Consistency in statistical inference and decision, (with discussion). J. Roy. Statist. Soc. Ser. B 23 1–25.
Smith, C. A. B. (1965)). Personal probability and statistical analysis, (with discussion). J. Roy. Statist. Soc. Ser. A 128 469–499.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Dempster, A.P. (2008). Upper and Lower Probabilities Induced by a Multivalued Mapping. In: Yager, R.R., Liu, L. (eds) Classic Works of the Dempster-Shafer Theory of Belief Functions. Studies in Fuzziness and Soft Computing, vol 219. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-44792-4_3
Download citation
DOI: https://doi.org/10.1007/978-3-540-44792-4_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-25381-5
Online ISBN: 978-3-540-44792-4
eBook Packages: EngineeringEngineering (R0)