Standard and nonstandard representability of positive uncertainty orderings

Andrea Capotorti; Giulianella Coletti; Barbara Vantaggi

Kybernetika (2014)

  • Volume: 50, Issue: 2, page 189-215
  • ISSN: 0023-5954

Abstract

top
Axioms are given for positive comparative probabilities and plausibilities defined either on Boolean algebras or on arbitrary sets of events. These axioms allow to characterize binary relations representable by either standard or nonstandard measures (i. e. taking values either on the real field or on a hyperreal field). We also study relations between conditional events induced by preferences on conditional acts.

How to cite

top

Capotorti, Andrea, Coletti, Giulianella, and Vantaggi, Barbara. "Standard and nonstandard representability of positive uncertainty orderings." Kybernetika 50.2 (2014): 189-215. <http://eudml.org/doc/261860>.

@article{Capotorti2014,
abstract = {Axioms are given for positive comparative probabilities and plausibilities defined either on Boolean algebras or on arbitrary sets of events. These axioms allow to characterize binary relations representable by either standard or nonstandard measures (i. e. taking values either on the real field or on a hyperreal field). We also study relations between conditional events induced by preferences on conditional acts.},
author = {Capotorti, Andrea, Coletti, Giulianella, Vantaggi, Barbara},
journal = {Kybernetika},
keywords = {comparative probability; comparative plausibilities; hyperreal field; representability by nonstandard measures; comparative probability; comparative plausibilities; hyperreal field; representability by nonstandard measures},
language = {eng},
number = {2},
pages = {189-215},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Standard and nonstandard representability of positive uncertainty orderings},
url = {http://eudml.org/doc/261860},
volume = {50},
year = {2014},
}

TY - JOUR
AU - Capotorti, Andrea
AU - Coletti, Giulianella
AU - Vantaggi, Barbara
TI - Standard and nonstandard representability of positive uncertainty orderings
JO - Kybernetika
PY - 2014
PB - Institute of Information Theory and Automation AS CR
VL - 50
IS - 2
SP - 189
EP - 215
AB - Axioms are given for positive comparative probabilities and plausibilities defined either on Boolean algebras or on arbitrary sets of events. These axioms allow to characterize binary relations representable by either standard or nonstandard measures (i. e. taking values either on the real field or on a hyperreal field). We also study relations between conditional events induced by preferences on conditional acts.
LA - eng
KW - comparative probability; comparative plausibilities; hyperreal field; representability by nonstandard measures; comparative probability; comparative plausibilities; hyperreal field; representability by nonstandard measures
UR - http://eudml.org/doc/261860
ER -

References

top
  1. Yaghlane, B. Ben, Smets, P., Mellouli, K., 10.1007/3-540-44652-4_30, Lect. Notes in Comput. Sci. 2143 (2001), 340-349. MR1909832DOI10.1007/3-540-44652-4_30
  2. Bernardi, S., Coletti, G., 10.1007/3-540-44652-4_11, Lect. Notes in Comput. Sci. 2143 (2001), 108-119. Zbl1001.68531DOI10.1007/3-540-44652-4_11
  3. Blume, L., Brandenburger, A., Dekel, E., 10.2307/2938240, Econometrica 59 (1991), 1, 61-79. Zbl0732.90005MR1085584DOI10.2307/2938240
  4. Capotorti, A., Coletti, G., Vantaggi, B., Non-additive ordinal relations representable by lower or upper probabilities., Kybernetika 34 (1998), 10, 79-90. Zbl1274.68518MR1619057
  5. Capotorti, A., Coletti, G., Vantaggi, B., 10.1007/s11238-007-9052-4, Theory and Decision 64 (2008), 119-146. Zbl1136.91392MR2399934DOI10.1007/s11238-007-9052-4
  6. Chateauneuf, A., Jaffray, J. Y., 10.1016/0022-2496(84)90026-9, J. Math. Psychol. 28 (1984), 191-204. Zbl0558.60003MR0763783DOI10.1016/0022-2496(84)90026-9
  7. Coletti, G., 10.1016/0022-2496(90)90034-7, J. Math. Psychol. 34 (1990), 297-310. Zbl0713.60003MR1068441DOI10.1016/0022-2496(90)90034-7
  8. Coletti, G., 10.1109/21.328932, IEEE Tras. Systems, Man, and Cybernetics 24 (1994), 12, 1747-1754. MR1302033DOI10.1109/21.328932
  9. Coletti, G., Mastroleo, M., Conditional belief functions: a comparison among different definitions., In: Proc. 7th Workshop on Uncertainty Processing (WUPES), 2006. 
  10. Coletti, G., Scozzafava, R., 10.1002/int.20133, Internat. J. of Intelligent Systems 21 (2006), 229-259. Zbl1160.68582DOI10.1002/int.20133
  11. Coletti, G., Scozzafava, R., Vantaggi, B., 10.1007/978-3-642-02906-6_48, Lect. Notes in Computer Science LNAI 5590 (2009), 554-565. Zbl1245.62012MR2893315DOI10.1007/978-3-642-02906-6_48
  12. Coletti, G., Vantaggi, B., 10.1007/s11238-005-4570-4, Theory and Decision 60 (2006), 137-174. Zbl1119.91029MR2226911DOI10.1007/s11238-005-4570-4
  13. Coletti, G., Vantaggi, B., A view on conditional measures through local representability of binary relations., Internat. J. Approximate Reasoning 60 (2006), 137-174. Zbl1184.68500MR2226911
  14. Coletti, G., Vantaggi, B., Conditional not-additive measures and fuzzy sets., In: Proc. ISIPTA 2013, pp. 67-76. 
  15. Finetti, B. de, Sul significato soggettivo delle probabilità., Fundam. Mat. 17 (1931), 293-329. 
  16. Finetti, B. de, La prevision: Ses lois logiques, ses sources subjectives., Ann. Inst. Henri Poincaré, Section B 7 (1937), l-68. Zbl0017.07602
  17. Finetti, B. de, Teoria della Probabilità., Einaudi, Torino 1970 (Engl. transl.) Theory of Probability, Wiley and Sons, London 1974. 
  18. Dempster, A. P., 10.1214/aoms/1177698950, Ann. Math. Statist. 38 (1967), 325-339. Zbl0168.17501MR0207001DOI10.1214/aoms/1177698950
  19. Dempster, A. P., A generalization of Bayesian inference., The Royal Stat. Soc. B 50 (1968), 205-247. Zbl0169.21301MR0238428
  20. Denoeux, T., Smets, P., 10.1109/TSMCB.2006.877795, IEEE Trans. on Systems, Man and Cybernetics B 36 (2006), 6, 1395-1406. DOI10.1109/TSMCB.2006.877795
  21. Dubins, L. E., 10.1214/aop/1176996451, Ann. Probab. 3 (1975), 89-99. MR0358891DOI10.1214/aop/1176996451
  22. Dubois, D., Fargier, H., Vantaggi, B., An axiomatization of conditional possibilistic preference functionals., Lect. Notes LNAI 4724 (2007), 803-815. Zbl1148.68511
  23. Fenchel, W., Convex Cones Sets and Functions., Lectures at Princeton University, Princeton 1951. Zbl0053.12203
  24. Fagin, R., Halpern, J. Y., Megido, N., 10.1016/0890-5401(90)90060-U, Information and Computation 87 (1990), 78-128. MR1055950DOI10.1016/0890-5401(90)90060-U
  25. Halpern, J .Y., 10.1016/j.geb.2009.03.013, Games and Economic Behavior 68 (2010), 1, 155-179. Zbl1208.60005MR2577384DOI10.1016/j.geb.2009.03.013
  26. Ghirardato, P., 10.1007/s001990100188, Economic Theory 20 (2002), 83-92. Zbl1030.91017MR1920674DOI10.1007/s001990100188
  27. Holzer, S., On coherence and conditional prevision., Bollettino UMI, Serie VI-C IV (1985), 1, 441-460. Zbl0584.60001MR0805231
  28. Jaffray, J. Y., 10.1109/21.179852, IEEE Trans. on Systems, Man, and Cybernetics 22 (1992), 1144-1152. Zbl0769.62001MR1202571DOI10.1109/21.179852
  29. Koopman, B. O., 10.2307/1969003, Ann. Math. 41 (1940), 269-292. Zbl0024.05001MR0001474DOI10.2307/1969003
  30. Kraft, C., Pratt, J., Seidenberg, A., 10.1214/aoms/1177706260, Ann. Math. Statist. 30 (1959), 408-419. Zbl0173.19606MR0102850DOI10.1214/aoms/1177706260
  31. Krauss, P. H., Representation of conditional probability measures on Boolean algebras., Acta Mathematica Academiae Sceintiarum Hungaricae 19 (1068), 3-4, 229-241. Zbl0174.49001MR0236080
  32. Lehmann, D., Generalized qualitative probability: Savage revisited., In: Proc. UAI'96, pp. 381-388. MR1617222
  33. Narens, L., 10.1016/0022-2496(74)90030-3, J. Math. Psychol. 11 (1974), 404-430. Zbl0307.02038MR0363541DOI10.1016/0022-2496(74)90030-3
  34. Paris, J., A note on the Dutch Book method., In: Proc. Second International Symposium on Imprecise Probabilities and their Applications (G. De Cooman, T. Fine, and T. Seidenfeld, eds.), ISIPTA 2001, Shaker Publishing Company, Ithaca, pp. 301-306. 
  35. Rényi, A., 10.1137/1101005, Theor. Probab. Appl. 1 (1956), 61-71. Zbl0073.12302MR0085639DOI10.1137/1101005
  36. Regazzini, E., 10.1007/BF02924866, Rendiconti Sem. Mat. Fis. Milano 55 (1985), 69-89. Zbl0683.60005MR0933711DOI10.1007/BF02924866
  37. Regoli, G., Rational comparisons and numerical representation., In: Decision Theory and Decision Analysis: Trends and Challenges, Academic Press, New York 1994. 
  38. Robinson, A., Non-Standard Analysis., North Holland, Amsterdam 1966. Zbl0843.26012MR0205854
  39. Savage, L. J., The Foundations of Statistics., Wiley, New York 1954. Zbl0276.62006MR0063582
  40. Shafer, G., 10.1214/aop/1176994941, Ann. Probab. 7 (1979), 827-839. Zbl0414.60002MR0542132DOI10.1214/aop/1176994941
  41. Vantaggi, B., 10.1016/j.mathsocsci.2010.06.002, Math. Soc. Sci. 60 (2010), 104-112. Zbl1232.91160MR2663973DOI10.1016/j.mathsocsci.2010.06.002
  42. Walley, P., 10.1214/aos/1176350603, Ann. Statist. 4 (1987), 1439-1465. Zbl0645.62003MR0913567DOI10.1214/aos/1176350603
  43. Walley, P., Statistical Reasoning with Imprecise Probabilities., Chapman and Hall, London 1991. Zbl0732.62004MR1145491
  44. Williams, P. M., Notes on Conditional Previsions., Working Paper School of Mathematical and Physical Sciences, The University of Sussex, 1975. Zbl1114.60005MR2295423
  45. Wong, S. K. M., Tao, Y. Y., Bollmann, P., Burger, H. C., 10.1109/21.108290, IEEE Trans. Systems, Man, and Cybernet. 21 (1991), 726-734. MR1143669DOI10.1109/21.108290

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.