Servicios Personalizados
Revista
Articulo
Indicadores
- Citado por SciELO
- Accesos
Links relacionados
- Similares en SciELO
Compartir
Polibits
versión On-line ISSN 1870-9044
Polibits no.42 México jul./dic. 2010
Expected Utility from Multinomial Secondorder Probability Distributions
David Sundgren
University of Gävle, Sweden. (dsn@hig.se).
Manuscript received June 8, 2010.
Manuscript accepted for publication July 25, 2010.
Abstract
We consider the problem of maximizing expected utility when utilities and probabilities are given by discrete probability distributions so that expected utility is a discrete stochastic variable. As for discrete secondorder distributions, that is probability distributions where the variables are themselves probabilities, the multinomial family is a reasonable choice at least if firstorder probabilities are interpreted as relative frequencies. We suggest a decision rule that reflects the uncertainty present in distributionbased probabilities and utilities and we show an example of this rule in action with multinomial secondorder distributions.
Key words: Imprecise probability. secondorder probability, discrete probability distributions, multinomial distributions, expected utilty.
DESCARGAR ARTÍCULO EN FORMATO PDF
REFERENCES
[1] A. P. Dempster, "Upper and lower probabilities induced by a multivalued mapping," Annals of Mathematical Statistics, vol xxxviii, pp. 325399, 1967. [ Links ]
[2] G. Shafer, A Mathematical theory of evidence. Princeton University Press, 1976. [ Links ]
[3] P. Huber, "The case of choquet capacities in statistics," Bulletin of the International Statistical Institute, 45, pp. 181188, 1973. [ Links ]
[4] P. Huber and V. Strassen, "Minimax tests and the neymanpearsons lemma for capacities," Annals of Statistics, 1, pp. 251263, 1973. [ Links ]
[5] I. Good, "Subjective probability as the measure of a nonmeasurable set," in Logic, Methodology, and the Philosophy of Science, P. Suppes, B. Nagel, and A. Tarski, Eds. Stanford University Press, 1962, pp. 319329. [ Links ]
[6] R. F. Nau, "Uncertainty aversion with secondorder utilities and probabilities," Management Science, vol. 52, no. 1, pp. 136145, 2006. [ Links ]
[7] L. Ekenberg and J. Thorbiornson, "Secondorder decision analysis," International Journal of Uncertainty, Fuzziness and KnowledgeBased Systems, vol. 9, No 1, vol. 9, no. 1, pp. 1338, 2001. [ Links ]
[8] L. V. Utkin and T. Augustin, "Decision making with imprecise secondorder probabilities," in ISIPTA '03 Proceedings of the Third International Symposium on Imprecise Probabilities and Their Applications, 2003, pp. 547561. [ Links ]
[9] L. V. Utkin, "Imprecise secondorder hierarchical uncertainty model," International Journal of Uncertainty, Fuzziness and KnowledgeBased Systems, Volume 11:3, pp. 301317, 2003. [ Links ]
[10] G. Choquet, "Theory of capacities," Ann. Inst. Fourier, pp. 5:131295, 1954. [ Links ]
[11] C. A. B. Smith, "Consistency in statistical inference and decision," Journal ofthe Royal Statistical Society, Series B, xxiii, pp. 125, 1961. [ Links ]
[12] P. Walley, Statistical reasoning with Imprecise Probabilities. Chapman and Hall, 1991. [ Links ]
[13] F. Cozman, "A brief introduction to the theory of sets of probability measures," cMU Tech. report CMURITR 9724. [ Links ]
[14] J. K. Satia and R. E. L. Jr., "Markovian decision processes with uncertain transition probabilities," Operations Research, vol. 21, pp. 728740, 1973. [ Links ]
[15] H. Robbins, "Asymptotically subminimax solutions to compound statistical decision problems," Proc. Second Berkerly Symp. Math. Stat. Probab., vol. 1, 1951. [ Links ]
[16] I. J. Good, "Rational decisions," Journal of the Royal Statistical Society, Series B, vol. 14 (1), pp. 107114, 1952. [ Links ]
[17] I. Levi, The Enterprise of Knowledge. MIT Press, 1980. [ Links ]
[18] M. Zaffalon, K. Wesnes, and O. Petrini, "Reliable diagnoses of dementia by the naive credal classifier inferred from incomplete cognitive data," Artificial Intelligence in Medicine, vol. 29, no. 12, pp. 6179, 2003. [ Links ]
[19] M. C. Troffaes, "Decision making under uncertainty using imprecise probabilities," International Journal of Approximate Reasoning, vol. 45, no. 1, pp. 17 29, 2007. [ Links ]
[20] D. Sundgren, L. Ekenberg, and M. Danielson, "Some properties of aggregated distributions over expected values," in MICAI 2008: Advances in Artificial Intelligence, ser. Lecture Notes in Computer Science, no. 5317. Springer, 2008, pp. 699709. [ Links ]