Abstract
If we interpret the statistical likelihood function as a measure of the relative plausibility of the probabilistic models considered, then we obtain a hierarchical description of uncertain knowledge, offering a unified approach to the combination of probabilistic and possibilistic uncertainty. The fundamental advantage of the resulting fuzzy probabilities with respect to imprecise probabilities is the ability of using all the information provided by the data.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Buckley, J.J.: Fuzzy Probability and Statistics. Studies in Fuzziness and Soft Computing, vol. 196. Springer, New York (2006)
Cattaneo, M.: Statistical Decisions Based Directly on the Likelihood Function. PhD Thesis, ETH Zurich (2007), http://e-collection.ethz.ch
Dahl, F.A.: Representing human uncertainty by subjective likelihood estimates. Internat. J. Approx. Reason 39(1), 85–95 (2005)
De Cooman, G.: A behavioural model for vague probability assessments. Fuzzy Sets Syst. 154, 305–358 (2005)
Dubois, D.: Possibility theory and statistical reasoning. Comput. Stat. Data Anal. 51, 47–69 (2006)
Dubois, D., Moral, S., Prade, H.: A semantics for possibility theory based on likelihoods. J. Math. Anal. Appl. 205(2), 359–380 (1997)
Edwards, A.W.F.: Likelihood, 2nd edn. Johns Hopkins University Press, Baltimore (1992)
Fisher, R.A.: On the mathematical foundations of theoretical statistics. Philos. Trans. R. Soc. Lond., Ser. A 222, 309–368 (1922)
Gärdenfors, P., Sahlin, N.E.: Unreliable probabilities, risk taking, and decision making. Synthese 53, 361–386 (1982)
Hisdal, E.: Are grades of membership probabilities? Fuzzy Sets Syst. 25, 325–348 (1988)
Kullback, S., Leibler, R.A.: On information and sufficiency. Ann. Math. Stat. 22, 79–86 (1951)
Moral, S.: Calculating uncertainty intervals from conditional convex sets of probabilities. In: Dubois, D., Wellman, M.P. (eds.) Proceedings of Uncertainty in Artificial Intelligence (UAI 1992, Stanford, CA, USA), San Mateo, CA, USA, pp. 199–206. Morgan Kaufmann, San Francisco (1992)
Pawitan, Y.: In All Likelihood: Statistical Modelling and Inference Using Likelihood. Oxford University Press, New York (2001)
Viertl, R., Hareter, D.: Beschreibung und Analyse unscharfer Information. Springer, Wien (2006)
Walley, P.: Statistical Reasoning with Imprecise Probabilities. Monographs on Statistics and Applied Probability, vol. 42. Chapman and Hall, Ltd., London (1991)
Wilson, N.: Modified upper and lower probabilities based on imprecise likelihoods. In: De Cooman, G., Fine, T.L., Seidenfeld, T. (eds.) Proceedings of the 2nd International Symposium on Imprecise Probabilities and Their Applications (ISIPTA 2001, Ithaca, USA), pp. 370–378. Shaker Publishing, Maastricht (2001)
Zadeh, L.A.: Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst. 1(1), 3–28 (1978)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Cattaneo, M.E.G.V. (2008). Fuzzy Probabilities Based on the Likelihood Function. In: Dubois, D., Lubiano, M.A., Prade, H., Gil, M.Á., Grzegorzewski, P., Hryniewicz, O. (eds) Soft Methods for Handling Variability and Imprecision. Advances in Soft Computing, vol 48. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85027-4_6
Download citation
DOI: https://doi.org/10.1007/978-3-540-85027-4_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-85026-7
Online ISBN: 978-3-540-85027-4
eBook Packages: EngineeringEngineering (R0)