Formal Representations of Uncertainty

Decision-making Process: Concepts and Methods

Volumn , Issue , 2010, Pages 85-156

  Dubois, Didier ^a   Prade, Henri ^a  

^a UNIVERSITÉ PAUL SABATIER   (France)

Author keywords

Economic decision problems; Empirical observation; Formalizing uncertainty; Probabilistic representations; Uncertainty theories

Indexed keywords

EID: 84885702167     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1002/9780470611876.ch3     Document Type: Chapter

References (133)

1. ADAMS E., LEVINE H., "On the uncertainties transmitted from premises to conclusions in deductive inferences", Synthese, vol. 30, p. 429-460, 1975.
2. BAUDRIT C., DUBOIS D., "Practical representations of incomplete probabilistic knowledge", Computational Statistics and Data Analysis, vol. 51, num. 1, p. 86-108, 2006.
3. BENFERHAT S., DUBOIS D., PRADE H., "Non-monotonic reasoning, conditional objects and possibility theory", Artificial Intelligence, vol. 92, p. 259-276, 1997.
4. BERNARD J.-M., "An introduction to the imprecise Dirichlet model for multinomial data", International Journal of Approximate Reasoning, vol. 39, num. 2-3, p. 123-150, 2005.
5. BIRNBAUM Z. W., "On random variables with comparable peakedness", Annals of Mathematical Statistics, vol. 19, p. 76-81, 1948.
6. BOOLE G., An Investigation of the Laws of Thought on which are Founded the Mathematical Theory of Logic and Probabilities, MacMillan, reprinted by Dover, New York, 1958.
7. BUDESCU D., WALLSTEIN T., "Processing linguistic probabilities: general principles and empirical evidence", BUSEMEYER J., HASTIE R., MEDIN D., Eds., The Psychology of Learning and Motivation: Decision-Making from the Perspective of Cognitive Psychology, p. 275-318, Academic Press, 1995.
8. CALABRESE P., "An algebraic synthesis of the foundations of logic and probability", Information Sciences, vol. 42, p. 187-237, 1987.
9. CALVO T., MAYOR G., MESIAR R., Eds., Aggregation Operators: New Trends and Applications, vol. 97 of Studies in Fuzziness and Soft Computing, Physica-Verlag, Heidelberg, 2002.
10. DE CAMPOS L. M., HUETE J., MORAL S., "Probability intervals: a tool for uncertain reasoning", International Journal of Uncertainty Fuzziness and Knowledge-Based Systems, vol. 2, p. 167-196, 1994.
11. CHANAS S., NOWAKOWSKI M., "Single value simulation of fuzzy variable", Fuzzy Sets and Systems, vol. 25, p. 43-57, 1988.
12. CHATEAUNEUF A., JAFFRAY J., "Some characterizations of lower probabilities and other monotone capacities through the use of Moebius inversion", Mathematical Social Sciences, vol. 17, p. 263-283, 1989.
13. CHATEAUNEUF A., "Combination of compatible belief functions and relation of specificity", FEDRIZZI M., KACPRZYK J., YAGER R., Eds., Advances in the Dempster- Shafer Theory of Evidence, p. 98-114, Wiley, New York, 1994.
14. CHATEAUNEUF A., "Decomposable capacities, distorted probabilities and concave capacities", Mathematical Social Sciences, vol. 31, p. 19-37, 1996.
15. CHELLAS B., Modal Logic, Cambridge University Press, Cambridge, UK, 1978.
16. CHOQUET G., "Theory of capacities", Annales de l'Institut Fourier, vol. 5, num. 4, p. 131-295, 1953.
17. COLETTI G., SCOZZAFAVA R., Probabilistic Logic in a Coherent Setting, Kluwer Academic Pub, 2002.
18. COLETTI G., SCOZZAFAVA R., "Coherent conditional probability as a measure of uncertainty of the relevant conditioning events", Proceedings of ECSQARU03, LNAI 2711, Aalborg, Springer Verlag, p. 407-418, 2003.
19. COOKE R. M., Experts in Uncertainty, Oxford University Press, Oxford, UK, 1991.
20. DE COOMAN G., "A behavioural model for vague probability assessments", Fuzzy Sets and Systems, vol. 154, num. 3, p. 305-358, 2005.
21. DE COOMAN G., AEYELS D., "Supremum-preserving upper probabilities", Information Sciences, vol. 118, p. 173-212, 1999.
22. DE FINETTI B., "La logique de la probabilité", Actes du Congrès Inter. de Philosophie Scientifique (1935), Paris, France, Hermann et Cie Editions, p. IV1-IV9, 1936.
23. DE FINETTI B., "La prévision: ses lois logiques, ses sources subjectives", Annales de l'Institut Poincaré, vol. 7, p. 1-68, 1937.
24. DE FINETTI B., Theory of Probability, Wiley, New York, 1974.
25. DEMPSTER A. P., "Upper and lower probabilities induced by a multivalued mapping", Annals of Mathematical Statistics, vol. 38, p. 325-339, 1967.
26. DENOEUX T., "Conjunctive and disjunctive combination of belief functions induced by nondistinct bodies of evidence", Artifical Intelligence, vol. 172, num. 2-3, p. 234-264, 2008.
27. DESTERCKE S., DUBOIS D., CHOJNACKI E., "Cautious conjunctive merging of belief functions", MELLOULI K., Ed., Symbolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU), vol. 4724 of Lecture Notes in Artificial Intelligence, LNAI, http://www.springerlink.com, Springer, p. 332-343, 2007.
28. DESTERCKE S., DUBOIS D., CHOJNACKI E., "Possibilistic information fusion using maximal coherent subsets", IEEE International Conference on Fuzzy Systems, London, UK, http://www.ieee.org/, IEEE, p. 445-450, 2007.
29. DESTERCKE S., DUBOIS D., CHOJNACKI E., "Relating practical representations of imprecise probabilities", DE COOMAN G., VEJNAROVA J., ZAFFALON M., Eds., International Symposium on Imprecise Probability: Theories and Applications (ISIPTA), Prague (Czech Republic), http://www.sipta.org, SIPTA, p. 155-164, 2007.
30. DOMOTOR Z., "Probability kinematics: conditional and entropy principles", Synthese, vol. 63, p. 74-115, 1985.
31. DUBOIS D., PRADE H., "0n several representations of an uncertain body of evidence", GUPTA M., SANCHEZ E., Eds., Fuzzy Information and Decision Processes, p. 167-181, North-Holland, Amsterdam, 1982.
32. DUBOIS D., PRADE H., "Unfair coins and necessity measures: a possibilistic interpretation of histograms", Fuzzy Sets and Systems, vol. 10, num. 1, p. 15-20, 1983.
33. DUBOIS D., "Belief structures, possibility theory and decomposable confidence measures on finite sets", Computers and Artificial Intelligence (Bratislava), vol. 5, p. 403- 416, 1986.
34. DUBOIS D., PRADE H., "A set-theoretic view of belief functions: logical operations and approximations by fuzzy sets", International Journal of General Systems, vol. 12, p. 193-226, 1986.
35. DUBOIS D., PRADE H., "Properties of information measures in evidence and possibility theories", Fuzzy Sets and Systems, vol. 24, p. 161-182, 1987.
36. DUBOIS D., PRADE H., Possibility Theory, Plenum Press, New York, 1988.
37. DUBOIS D., PRADE H., "Representation and combination of uncertainty with belief functions and possibility measures", Computational Intelligence, vol. 4, num. 4, p. 244-264, 1988.
38. DUBOIS D., PRADE H., "Aggregation of possibility measures", KACPRZYK J., FEDRIZZI M., Eds., Multiperson Decision Making using Fuzzy Sets and Possibility Theory, p. 55-63, Kluwer, Dordrecht, 1990.
39. DUBOIS D., PRADE H., "Consonant approximations of belief functions", International Journal of Approximate Reasoning, vol. 4, num. 5/6, p. 419-449, 1990.
40. DUBOIS D., PRADE H., "Random sets and fuzzy interval analysis", Fuzzy Sets and Systems, vol. 42, p. 87-101, 1991.
41. DUBOIS D., PRADE H., "When upper probabilities are possibility measures", Fuzzy Sets and Systems, vol. 49, p. 65-74, 1992.
42. DUBOIS D., YAGER R., "Fuzzy set connectives as combinations of belief structures", Information Sciences, vol. 66, p. 245-275, 1992.
43. DUBOIS D., PRADE H., SANDRI S., "On possibility/probability transformations", LOWEN R., ROUBENS M., Eds., Fuzzy Logic. State of the Art, p. 103-112, Kluwer Academic Publishing, Dordrecht, 1993.
44. DUBOIS D., RAMER A., "Extremal properties of belief measures in the theory of evidence", International Journal of Uncertainty Fuzziness and Knowledge-Based Systems, vol. 1, num. 1, p. 57-68, 1993.
45. DUBOIS D., PRADE H., "Conditional objects as non-monotonic consequence relationships", Special issue on Conditional Event Algebra IEEE Transactions on Systems Man and Cybernetics, vol. 24, num. 12, p. 1724-1740, 1994.
46. DUBOIS D., PRADE H., "Possibility theory and data fusion in poorly informed environments", Control Engineering Practice, vol. 2, num. 5, p. 811-823, 1994.
47. DUBOIS D., PRADE H., SMETS P., "Representing partial ignorance", IEEE Transactions on Systems Man and Cybernetics, vol. 26, num. 3, p. 361-377, 1996.
48. DUBOIS D., MORAL S., PRADE H., "A semantics for possibility theory based on likelihoods", Journal of Mathematical Analysis and Applications, vol. 205, p. 359-380, 1997.
49. DUBOIS D.,MORAL S., PRADE H., "Belief change rules in ordinal and numerical uncertainty theories", DUBOIS D., PRADE H., Eds., Belief Change, p. 311-392, Kluwer Academic Publishing, 1998.
50. DUBOIS D., PRADE H., "Possibility theory: qualitative and quantitative aspects", SMETS P., Ed., Handbook on Defeasible Reasoning and Uncertainty Management Systems. Volume 1: Quantified Representation of Uncertainty and Imprecision, p. 169-226, Kluwer Academic Publ., Dordrecht, The Netherlands, 1998.
51. DUBOIS D., NGUYEN H. T., PRADE H., "Possibility theory, probability and fuzzy sets: misunderstandings, bridges and gaps", DUBOIS D., PRADE H., Eds., Fundamentals of Fuzzy Sets, The Handbooks of Fuzzy Sets Series, p. 343-438, Kluwer, Boston, Mass., 2000.
52. DUBOIS D., PRADE H., "Possibility theory in information fusion", DELLA RICCIA G., LENZ H., KRUSE R., Eds., Data Fusion and Perception, Vol. 431 in the CISM Courses and Lectures, p. 53-76, Springer-Verlag, Berlin, 2001.
53. DUBOIS D., PRADE H., SMETS P., "New semantics for quantitative possibility theory", ISIPTA '01, Proceedings of the Second International Symposium on Imprecise Probabilities and their Applications, Ithaca, NY, USA, Shaker, p. 152-161, 2001.
54. DUBOIS D., PRADE H., SMETS P., "A definition of subjective possibility", Badania Pperacyjne i Decyzje (Operation Research and Decisions), vol. 4, p. 7-22, 2003, Revised version in International Journal of Approximate Reasoning, vol. 48(2), p. 352-364, 2008.
55. DUBOIS D., FARGIER H., PRADE H., "Ordinal and probabilistic representations of acceptance", Journal of Artificial Intelligence Research (JAIR), vol. 22, p. 23-56, 2004.
56. DUBOIS D., FOULLOY L., MAURIS G., PRADE H., "Possibility/probability transformations, triangular fuzzy sets, and probabilistic inequalities", Reliable Computing, vol. 10, p. 273-297, 2004.
57. DUBOIS D., HUELLERMEIER E., "Comparing probability measures using possibility theory: a notion of relative peakedness", International Journal of Approximate Reasoning, vol. 45, p. 364-385, 2007.
58. FERSON S., GINZBURG L. R., "Different methods are needed to propagate ignorance and variability", Reliability Engineering and System Safety, vol. 54, p. 133-144, 1996.
59. FERSON S., GINZBURG L., KREINOVICH V., MYERS D., SENTZ K., Construction probability boxes and Dempster-Shafer structures, Report num. SANDD2002-4015, Sandia National Laboratories, 2003.
60. FINE T., Theories of Probability, Academic Press, New York, 1983.
61. FODOR J., YAGER R., "Fuzzy set-theoretic operators and quantifiers", DUBOIS D., PRADE H., Eds., Fundamentals of Fuzzy Sets, The Handbook of Fuzzy Sets Series, p. 125- 193, Kluwer Academic Publ., Dordrecht, 2000.
62. FRENCH S., "Group consensus probability disributions: a critical survey", Bayesian Statistics 2, p. 183-202, Elsevier Science, 1985.
63. FRIEDMAN N., HALPERN J., "Plausibility measures and default reasoning", Proceedings of the 13th National Conference on Artificial Intelligence, Portland, OR, p. 1297-1304, 1996.
64. GÄRDENFORS P., Knowledge in Flux: Modeling the Dynamics of Epistemic States, MIT Press, Cambridge, Mass, 1988.
65. GILES R., "Foundations for a theory of possibility", GUPTA M., SANCHEZ E., Eds., Fuzzy Information and Decision Processes, p. 1836-195, North-Holland, 1982.
66. GILBOA I., SCHMEIDLER D., "Maxmin expected utility with a non-unique prior", Journal of Mathematical Economics, vol. 18, p. 141-153, 1989.
67. GILBOA I., SCHMEIDLER D., "Updating ambiguous beliefs", MOSES Y., Ed., Theoretical Aspects of Reasoning About Knowledge (Proceedings of the 4th Conference TARK'92), p. 143-162, 1992.
68. GILBOA I., SCHMEIDLER D., "Case-based knowledge and induction", IEEE Transactions on Systems Man and Cybernetics Part A, vol. 30, num. 2, p. 85-95, 2000.
69. GOOD I., "Subjective Probability as the measure of a non-measurable set", NAGEL E., ET AL., Eds., Logic, Probability and Philosophy of Science, Stanford University Press, Stanford, 1962.
70. GOODMAN I., NGUYEN H., WALKER E., Conditional Inference and Logic for Intelligent Systems: A Theory of Measure-Free Conditioning, North-Holland, Amsterdam, 1991.
71. HACKING I., The Emergence of Probability, Cambridge University Press, Cambridge, UK, 1975.
72. HALPERN J., "Defining relative likelihood in partially-ordered preferential structures", Journal of Artifical Intelligence Research, vol. 7, p. 1-24, 1997.
73. HINTIKKA J., Knowledge and Belief, Cornell University Press, Ithaca, NY, 1962.
74. JAFFRAY J., "Bayesian updating and belief functions", IEEE Transactions on Systems Man and Cybernetics, vol. 22, p. 1144-1152, 1992.
75. JAYNES E., "Where do we stand on maximum entropy", LEVINE I., TRIBUS M., Eds., The Maximum Entropy Formalism, MIT Press, Cambridge USA, 1979.
76. KAHNEMANN D., SLOVIC P., TVERSKY A., Eds., Judgment Under Uncertainty: Heuristics and Biases, Cambridge University Press, Cambridge, UK, 1979.
77. KLEMENT E., MESIAR R., PAP E., Triangular Norms, Kluwer Academic Pub., Boston, 2000.
78. KLIR G., PARVIZ B., "Probability-possibility transformation: a comparison", International Journal of General Systems, vol. 21, p. 291-310, 1992.
79. KOOPMAN B., "The bases of probability", Bulletin of American Mathematical Society, vol. 46, p. 763-774, 1940.
80. KRAFT C., PRATT J., SEIDENBERG A., "Intuitive probability on finite sets", Annals of Mathematical Statistics, vol. 30, p. 408-419, 1959.
81. KRIEGLER E., HELD H., "Utilizing belief functions for the estimation of future climate change", International Journal of Approximate Reasoning, vol. 39, p. 185-209, 2005.
82. KYBURG H., The Logical Foundations of Statistical Inference, Reidel, Dordrecht, The Netherlands, 1974.
83. LEHRER K., WAGNER C., Rational Consensus in Science and Society, Reidel, Dordrecht, 1981.
84. LEVI I., The Enterprise of Knowledge, The MIT Press, Cambridge, Mass., 1980.
85. LEWIS D., "Probabilities of conditionals and conditional probabilities", The Philosophical Review, vol. 85, p. 297-315, 1976.
86. LEWIS D., Counterfactuals, Basil Blackwell (1973), 2nd edition, Billing and Sons Ltd., Worcester, UK, 1986.
87. LINDLEY D., "Scoring rules and the inevitability of probability", International Statistical Review, vol. 50, p. 1-26, 1982.
88. LUCE R., RAIFFA H., Games and Decisions, Wiley, New York, 1981.
89. MARSCHAK J., "Personal probabilities of probabilities", Theory and Decision, vol. 6, p. 121-125, 1975.
90. MIRANDA E., COUSO I., GIL P., "Relationships between possibility measures and nested random sets", International Journal of Uncertainty Fuzziness and Knowledge-Based Systems, vol. 10, num. 1, p. 1-15, 2002.
91. MIRANDA E., COUSO I., GIL P., "A random set characterization of possibility measures", Information Science, vol. 168, num. 1-4, p. 51-75, 2004.
92. MOORE R. E., Interval Analysis, Prentice Hall, Englewood Cliffs, NJ, 1966.
93. MUNDICI D., "The logic of Ulam games with lies", BICCHIERI C., DALLA CHIARA M., Eds., Knowledge: Belief and Strategic Interaction, p. 275-284, Cambridge Univ. Press, 1992.
94. NEUMAIER A., "Clouds, fuzzy sets and probability intervals", Reliable Computing, vol. 10, p. 249-272, 2004.
95. PARIS J., The Uncertain Reasoner's Companion, Cambridge University Press, 1994.
96. PAWLAK Z., Rough Sets: Theoretical Aspects of Reasoning about Data, Kluwer Academic Publ., Dordrecht, 1991.
97. RAMSEY F. P., "Truth and probability", KYBURG J., SMOKLER H., Eds., Studies in Subjective Probability, p. 23-52, Krieger Pub. Co, Huntington, NY, 1980.
98. RAUFASTE E., DA SILVA NEVES R., MARINÉ C., "Testing the descriptive validity of possibility theory in human judgments of uncertainty", Artificial Intelligence, vol. 149, p. 197-218, 2003.
99. RESCHER N., MANOR R., "On inference from inconsistent premises", Theory and Decision, vol. 1, p. 179-219, 1970.
100. SAVAGE L., The Foundations of Statistics, Wiley (1954); 2nd edition, Dover Publications Inc., New York, 1972.
101. SCHMEIDLER D., "Subjective probability and expected utility without additivity", Econometrica, vol. 57, p. 571-587, 1989.
102. SHAPLEY L. S., "A value for n-person games", KUHN, TUCKER, Eds., Contributions to the Theory of Games, II, p. 307-317, Princeton University Press, 1953.
103. SHACKLE G. L., Decision, Order and Time in Human Affairs, (2nd edition), Cambridge University Press, UK, 1961.
104. SHAFER G., A Mathematical Theory of Evidence, Princeton University Press, 1976.
105. SHAFER G., "Non-additive probabilities in the work of Bernoulli and Lambert", Archive for History of Exact Sciences, vol. 19, p. 309-370, 1978.
106. SHAFER G., "Allocations of probability", Annals of Probability, vol. 7, p. 827-839, 1979.
107. SMARANDACHE F., DEZERT J., "Proportional conflict redistribution rules for information fusion", SMARANDACHE F. D. J., Ed., Applications and Advances of DSmT for Information Fusion. Vol. 2., p. 3-68, American Research Press, Rehoboth, N.M., 2006.
108. SMETS P., "Possibilistic inference from statistical data", Proceedings of the 2nd World Conference on Mathematics at the Service of Man, Las Palmas (Canary Islands), Spain, p. 611-613, Jun-Jul 1982.
109. SMETS P., "Constructing the pignistic probability function in a context of uncertainty", HENRION M., et al., Eds., Uncertainty in Artificial Intelligence, p. 29-39, North-Holland, Amsterdam, 1990.
110. SMETS P., KENNES R., "The transferable belief model", Artificial Intelligence, vol. 66, p. 191-234, 1994.
111. SMETS P., "The normative representation of quantified beliefs by belief functions", Artificial Intelligence, vol. 92, p. 229-242, 1997.
112. SMETS P., "The transferable belief model for quantified belief representation", Handbook of Defeasible Reasoning and Uncertainty Management Systems, vol. 1, p. 267- 301, Kluwer Academic, Dordrecht, The Netherlands, 1998.
113. SMETS P., "Quantified possibility theory seen as an hypercautious transferable belief model", Rencontres Francophones sur les Logiques Floues et ses Applications (LFA 2000), La Rochelle, France, Cepadues-Editions, p. 343-353, Oct 2000.
114. SMETS P., "Belief functions on real numbers", International Journal of Approximate Reasoning, vol. 40, p. 181-223, 2005.
115. SMETS P., "Analyzing the combination of conflicting belief functions", Information Fusion, vol. 8, num. 4, p. 387-412, 2007.
116. SMITH C., "Consistency in statistical inference and decision", Journal of the Royal Statistical Society, vol. B-23, p. 1-37, 1961.
117. STRAT T., "Continuous belief functions for evidential reasoning", National Conference on Artificial Intelligence (AAAI-84), Los Altos, CA, W. Kaufmann, p. 308-313, 1984.
118. SUGENO M., "Fuzzy measures and fuzzy integral: a survey", GUPTA M., SARIDIS G., GAINES B., Eds., Fuzzy Automata and Decision Processes, p. 89-102, North-Holland, Amsterdam, 1977.
119. TELLER P., "Conditionalization, observation and change of preference", HARPER W., HOOKER C., Eds., Foundations of Probability Theory, Statistical Inference, and Statistical Theories of Science, vol. 1, p. 205-259, D. Reide, Los Altos, CA, 1984.
120. WALLEY P., FINE T., "Towards a frequentist theory of upper and lower probability", The Annals of Statistics, vol. 10, p. 741-761, 1982.
121. WALLEY P., Statistical Reasoning with Imprecise Probabilities, Chapman and Hall, 1991.
122. WILLIAMS P., "Bayesian conditionalization and the principle of minimum information", British Journal for the Philosophy of Sciences, vol. 31, p. 131-144, 1980.
123. WONG S., YAO Y., BOLLMAN P., BURGER H., "Axiomatization of qualitative belief structure", IEEE Transactions on Systems Man and Cybern, vol. 21, p. 726-734, 1993.
124. YAGER R., "An introduction to applications of possibility theory", Human Systems Management, vol. 3, p. 246-269, 1983.
125. YAGER R. R., "On the Dempster-Shafer framework and new combination rules", Information Sciences, vol. 41, num. 2, p. 93-137, 1987.
126. YAMADA K., "A new combination of evidence based on compromise", Fuzzy Sets and Systems, vol. 159, num. 13, p. 1689-1708, 2008.
127. ZADEH L., "Fuzzy sets", Information and Control, vol. 8, p. 338-353, 1965.
128. ZADEH L., "The concept of a linguistic variable and its application to approximate reasoning, Part I", Information Sciences, vol. 8, p. 199-249, 1975.
129. ZADEH L., "The concept of a linguistic variable and its application to approximate reasoning, Part II", Information Sciences, vol. 8, p. 301-357, 1975.
130. ZADEH L., "The concept of a linguistic variable and its application to approximate reasoning, Part III", Information Sciences, vol. 9, p. 43-80, 1975.
131. ZADEH L., "Fuzzy sets as a basis for a theory of possibility", Fuzzy Sets and Systems, vol. 1, p. 3-28, 1978.
132. ZADEH L. A., "Fuzzy probabilities", Information Processing and Management, vol. 20, num. 3, p. 363-372, 1984.
133. ZADEH L., "Toward a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic", Fuzzy Sets and Systems, vol. 90, p. 111-127, 1997.