A geometric and game-theoretic study of the conjunction of possibility measures. (English) Zbl 1360.68845
Summary: In this paper, we study the conjunction of possibility measures when they are interpreted as coherent upper probabilities, that is, as upper bounds for some set of probability measures. We identify conditions under which the minimum of two possibility measures remains a possibility measure. We provide graphical way to check these conditions, by means of a zero-sum game formulation of the problem. This also gives us a nice way to adjust the initial possibility measures so their minimum is guaranteed to be a possibility measure. Finally, we identify conditions under which the minimum of two possibility measures is a coherent upper probability, or in other words, conditions under which the minimum of two possibility measures is an exact upper bound for the intersection of the credal sets of those two possibility measures.
MSC:
| 68T37 | Reasoning under uncertainty in the context of artificial intelligence |
| 60A10 | Probabilistic measure theory |
| 91A80 | Applications of game theory |
Keywords:
possibility measure; conjunction; imprecise probability; game theory; natural extension; coherenceReferences:
| [1] | Alvarez, Diego A., A Monte Carlo-based method for the estimation of lower and upper probabilities of events using infinite random sets of indexable type, Fuzzy Sets Syst., 160, 3, 384-401 (2009) · Zbl 1178.60009 |
| [2] | Baudrit, Cédric; Guyonnet, Dominique; Dubois, Didier, Joint propagation of variability and imprecision in assessing the risk of groundwater contamination, J. Contam. Hydrol., 93, 1, 72-84 (2007) |
| [3] | Bemporad, Alberto; Fukuda, Komei; Torrisi, Fabio D., Convexity recognition of the union of polyhedra, Comput. Geom., 18, 3, 141-154 (2001), 10.1016/S0925-7721(01)00004-9 · Zbl 0976.68163 |
| [4] | Benavoli, Alessio; Antonucci, Alessandro, An aggregation framework based on coherent lower previsions: application to zadeh’s paradox and sensor networks, Int. J. Approx. Reason., 51, 9, 1014-1028 (2010) · Zbl 1348.68243 |
| [5] | Chateauneuf, Alain, Combination of compatible belief functions and relation of specificity, (Advances in the Dempster-Shafer Theory of Evidence (1994), Wiley), 97-114 |
| [8] | de Cooman, Gert, Possibility theory I: the measure- and integral-theoretic groundwork, Int. J. Gen Syst, 25, 291-323 (1997) · Zbl 0955.28012 |
| [9] | De Cooman, Gert, A behavioural model for vague probability assessments, Fuzzy Sets Syst., 154, 3, 305-358 (2005) · Zbl 1123.62006 |
| [10] | de Cooman, Gert; Aeyels, Dirk, Supremum preserving upper probabilities, Inf. Sci., 118, 173-212 (1999) · Zbl 0952.60009 |
| [11] | de Cooman, Gert; Troffaes, Matthias C. M., Coherent lower previsions in systems modelling: products and aggregation rules, Rel. Eng. Syst. Safety, 85, 1-3, 113-134 (2004) |
| [12] | Destercke, Sébastien; Dubois, Didier, Idempotent conjunctive combination of belief functions: extending the idempotent rule of possibility theory, Inf. Sci., 181, 3925-3945 (2011) · Zbl 1242.68324 |
| [13] | Destercke, Sébastien; Dubois, Didier; Chojnacki, Eric, A consonant approximation of the product of independent consonant random sets, Int. J. Uncertainty Fuzz. Knowl.-Based Syst., 17, 06, 773-792 (2009) · Zbl 1185.68706 |
| [14] | Dubois, Didier; Prade, Henri, A set theoretic view of belief functions, Int. J. Gen Syst, 12, 193-226 (1986) |
| [15] | Dubois, Didier; Prade, Henri, Possibility Theory - An Approach to Computerized Processing of Uncertainty (1988), Plenum Press: Plenum Press New York · Zbl 0703.68004 |
| [16] | Dubois, Didier; Prade, Henri, Representation and combination of uncertainty with belief functions and possibility measures, Comput. Intell., 4, 244-264 (1988) |
| [17] | Dubois, Didier; Prade, Henri, When upper probabilities are possibility measures, Fuzzy Sets Syst., 49, 1, 65-74 (1992) · Zbl 0754.60003 |
| [18] | Dubois, Didier; Prade, Henri; Yager, Ron, Merging fuzzy information, (Prade, H.; Bezdek, J. C.; Didier, D., Fuzzy Sets in Approximative Reasoning and Information Systems. Fuzzy Sets in Approximative Reasoning and Information Systems, Handbook of Fuzzy Sets (1999), Kluwer: Kluwer Dordrecht), 335-401, (Chapter 6) · Zbl 0951.94028 |
| [19] | Giles, Robin, Semantics for fuzzy reasoning, Int. J. Man Mach. Stud., 17, 4, 401-415 (1982), doi: 10.1016/S0020-7373(82)80041-2 · Zbl 0515.03013 |
| [20] | Hunter, Anthony; Liu, Weiru, A context-dependent algorithm for merging uncertain information in possibility theory, IEEE Trans. Syst. Man Cybern.: Part A, 38, 6, 1385-1397 (2008) |
| [21] | Kolmogorov, Andrei N., Foundations of the Theory of Probability (1950), Chelsea Publishing Company: Chelsea Publishing Company New York · Zbl 0037.08101 |
| [23] | Duncan Luce, R.; Raiffa, Howard, Games and Decisions: Introduction and Critical Survey (1957), Dover Publications · Zbl 0084.15704 |
| [24] | Miranda, Enrique, A survey of the theory of coherent lower previsions, Int. J. Approx. Reason., 48, 2, 628-658 (2008) · Zbl 1184.68519 |
| [25] | Miranda, Enrique; Couso, Inés; Gil, Pedro, Extreme points of credal sets generated by 2-alternating capacities, Int. J. Approx. Reason., 33, 1, 95-115 (2003) · Zbl 1029.68135 |
| [26] | Miranda, Enrique; Cooman, Gert De, Epistemic independence in numerical possibility theory, Int. J. Approx. Reason., 32, 1, 23-42 (2003) · Zbl 1033.68117 |
| [27] | Moral, Serafín; del Sagrado, José, Aggregation of imprecise probabilities, (Bouchon-Meunier, B., Aggregation and Fusion of Imperfect Information (1998), Physica-Verlag: Physica-Verlag New York), 162-188 |
| [28] | Palacios, Ana M.; Sánchez, Luciano; Couso, Inés, Diagnosis of dyslexia with low quality data with genetic fuzzy systems, Int. J. Approx. Reason., 51, 8, 993-1009 (2010) |
| [29] | Sandri, Sandra A.; Dubois, Didier; Kalfsbeek, Henk W., Elicitation, assessment, and pooling of expert judgments using possibility theory, IEEE Trans. Fuzzy Syst., 3, 3, 313-335 (1995) |
| [30] | Smith, Cedric A. B., Consistency in statistical inference and decision, J. R. Stat. Soc. B, 23, 1-37 (1961) · Zbl 0124.09603 |
| [31] | Troffaes, Matthias C. M., Generalising the conjunction rule for aggregating conflicting expert opinions, Int. J. Intell. Syst., 21, 3, 361-380 (2006) · Zbl 1160.68583 |
| [32] | Troffaes, Matthias C. M.; Cooman, Gert de, Lower Previsions (2014), Wiley · Zbl 1305.60007 |
| [33] | Troffaes, Matthias C. M.; Miranda, Enrique; Destercke, Sébastien, On the connection between probability boxes and possibility measures, Inf. Sci., 224, 88-108 (2013), arXiv:1103.5594 · Zbl 1293.60005 |
| [36] | Walley, Peter, Statistical Reasoning with Imprecise Probabilities (1991), Chapman and Hall: Chapman and Hall London · Zbl 0732.62004 |
| [37] | Walley, Peter, Measures of uncertainty in expert systems, Artif. Intell., 83, 1-58 (1996) · Zbl 1506.68157 |
| [38] | Yager, Ron, Entropy and specificity in a mathematical theory of evidence, Int. J. Gen Syst., 9, 249-260 (1983) · Zbl 0521.94008 |
| [39] | Zadeh, Lofti A., Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets Syst., 1, 3-28 (1978) · Zbl 0377.04002 |
| [40] | Zaffalon, Marco; Miranda, Enrique, Probability and time, Artif. Intell., 198, 1-51 (2013) · Zbl 1284.68558 |
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