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Bets and boundaries: Assigning probabilities to imprecisely specified events. (English) Zbl 1159.03309
Summary: Uncertainty and vagueness/imprecision are not the same: one can be certain about events described using vague predicates and about imprecisely specified events, just as one can be uncertain about precisely specified events. Exactly because of this, a question arises about how one ought to assign probabilities to imprecisely specified events in the case when no possible available evidence will eradicate the imprecision (because, say, of the limits of accuracy of a measuring device).
Modelling imprecision by rough sets over an approximation space presents an especially tractable case to help get one’s bearings. Two solutions present themselves: the first takes as upper and lower probabilities of the event \(X\) the (exact) probabilities assigned \(X\)’s upper and lower rough-set approximations; the second, motivated both by formal considerations and by a simple betting argument, is to treat \(X\)’s rough-set approximation as a conditional event and assign to it a point-valued (conditional) probability.
With rough sets over an approximation space we get a lot of good behaviour. For example, in the first construction mentioned the lower probabilities are \(n\)-monotone, for every \(n \in \mathbb{N}^{+}\). When we examine other models of approximation/imprecision/vagueness, and in particular, proximity spaces, we lose a lot of that good behaviour. In the literature there is not (even) agreement on the definition of upper and lower approximations for events (subsets) in the underlying domain. Betting considerations suggest one choice and, again, ways to assign upper and lower and point-valued probabilities, but nothing works well.

03B48 Probability and inductive logic
03E70 Nonclassical and second-order set theories
68T37 Reasoning under uncertainty in the context of artificial intelligence
Full Text: DOI
[1] Allam A.A., Bakeir M.Y., Abo-Tabl E.A.: ’New Approach for Basic Rough Set Concepts’. In: Ślezak, D., Wang, G., Szczuka, M., Düntsch, I., Yao, Y.Y.(eds) Rough Sets, Fuzzy Sets, Data Mining, and Granular Computing: 10th International Conference, RSFDGrC 2005, Regina, Canada, August 31 - September 3, 2005, Proceedings, Part I, LNAI 3641, pp. 64–73. Springer, Berlin & Heidelberg (2005) · Zbl 1134.68521
[2] Baroni P., Vicig P.: ’On the Conceptual Status of Belief Functions with Respect to Coherent Lower Probabilities’. In: Benferhat, S., Besnard, P.(eds) Symbolic and Quantitative Approaches to Reasoning with Uncertainty: 6th European Conference, ECSQARU 2001, Toulouse, France, September 19-21, 2001, Proceedings, LNAI 2143, pp. 328–339. Springer, Berlin & Heidelberg (2001) · Zbl 1001.68567
[3] Beaubouef T., Petry F.: ’Vagueness in Spatial Data: Rough Set and Egg-Yolk Approaches’. In: Monostori, L., Váncza, J., Ali, M.(eds) Engineering of Intelligent Systems: 14th International Conference on Industrial and Engineering Applications of Artificial Intelligence and Expert Systems, IEA/AIE 2001 Budapest, Hungary, June 4-7, 2001 Proceedings, LNAI 2070, pp. 367–373. Springer, Berlin & Heidelberg (2001) · Zbl 0981.68650
[4] Bell J.L.: ’Orthologic, Forcing, and the Manifestation of Attributes’. In: Chang, C.-T., Wicks, M.J. (eds) Southeast Asian Conference on Logic (Studies in Logic and Foundations of Mathematics, 111), pp. 13–36. North-Holland, Amsterdam (1983) · Zbl 0553.03040
[5] Bennett B.: ’What is a Forest? On the Vagueness of Certain Geographic Concepts’. Topoi 20(2), 189–201 (2001)
[6] Bittner, T., and B. Smith, ’A Unified Theory of Granularity, Vagueness and Approximation’, presented at the 5th International Conference on Spatial Information Theory, COSIT 2001, Morro Bay, California, USA, September 19-23, 2001. Available on-line at http://www.qrg.northwestern.edu/papers/Files/BittnerSmithSVUG01.pdf
[7] Bittner T., Stell J.G.: ’Vagueness and Rough Location’. GeoInformatica 6(2), 99–121 (2002) · Zbl 1035.68568
[8] Bloch I.: ’On links between mathematical morphology and rough sets’. Pattern Recognition 33(9), 1487–1496 (2000) · Zbl 02181325
[9] Cattaneo G.: ’Fuzzy events and fuzzy logics in classical information systems’. Journal of Mathematical Analysis and Applications 75(2), 523–548 (1980) · Zbl 0447.94062
[10] Cattaneo, G., ’Canonical embedding of an abstract quantum logic into the partial Baer?-ring of complex fuzzy events’, Fuzzy Sets and Systems 9 (1–3):179–198, 1983 · Zbl 0537.03046
[11] Cattaneo G.: ’Generalized Rough Sets (Preclusivity Fuzzy-Intuitionistic (BZ) Lattices)’. Studia Logica 58(1), 47–77 (1997) · Zbl 0864.03040
[12] Cattaneo G., Marino G.: ’Non-usual orthocomplementations on partially ordered sets and fuzziness’. Fuzzy Sets and Systems 25(1), 107–123 (1988) · Zbl 0631.06005
[13] Cattaneo G., Nisticó G.: ’Brouwer–Zadeh posets and three-valued Łukasiewicz posets’. Fuzzy Sets and Systems 33(2), 165–190 (1989) · Zbl 0682.03036
[14] Cheng, J.-X., andW.-L. Chen, ’Quasi-discrete Closure Space and Generalized Rough Approximate Space Based on Binary Relation’, IEEE Proceedings of the Third International Conference on Machine Learning and Cybernetics, Shanghai, 26-29 August 2004, vol. 5, 2004, pp. 2212–2216.
[15] Coletti G., Scozzafava R.: ’Toward a general theory of conditional beliefs’. International Journal of Intelligent Systems 21(3), 229–259 (2006) · Zbl 1160.68582
[16] de Finetti B.(1995). ’La Logique de la probabilité’, Actes du congréès international de philosophie scientifique, Fasc. IV, Paris: Hermann, 1936, pp. 31-39; English translation by R.B. Angell, ’The Logic of Probability’, Philosophical Studies 77 (1):181–190, 1995.
[17] Dummett, M., ’Wang’s Paradox’, Synthese 30 (3–4):301–324, 1975; reprinted in Dummett, Truth and Other Enigmas, London: Duckworth, 1978, pp. 248–268. · Zbl 0318.02006
[18] Fagin R., Halpern J.Y.: ’Uncertainty, Belief, and Probability’. Computational Intelligence 7(3), 160–173 (1991) · Zbl 0718.68066
[19] Frege, G., Grundgesetze der Arithmetik, begriffsshriftlich abgeleitet, Vol. II, Jena: Hermann Pohle, 1903; partial English translation in P.T. Geach and M. Black (eds.), Translations from the Philosophical Writings of Gottlob Frege (third edition), Oxford: Basil Blackwell, 1982.
[20] Goldblatt R.: ’A Semantic Analysis of Orthologic’. Journal of Philosophical Logic 3(1–2), 19–35 (1974) · Zbl 0278.02023
[21] Intan R., Mukaidono M.: ’A Proposal of Probability of Rough Event Based on Probability of Fuzzy Event’. In: Alpigini, J.J., Peters, J.F., Skowron, A., Zhong N, N.(eds) Rough Sets and Current Trends in Computing: Third International Conference, RSCTC 2002, Malvern, PA, USA, October 14-16, 2002, pp. 357–364. Proceedings, LNAI 2475. Berlin & Heidelberg (2002) · Zbl 1013.68563
[22] Iwiński T.: ’Algebraic approach to rough sets’. Bulletin of the Polish Academy of Sciences: Mathematics 35(9–10), 673–683 (1987) · Zbl 0639.68125
[23] Järvinen J.: ’Approximations and Rough Sets Based on Tolerances’. In: Ziarko, W., Yao, Y.Y. (eds) Rough Sets and Current Trends in Computing: Second International Conference, RSCTC 2000 Banff, Canada, October 16-19, 2000 Revised Papers, LNAI 2005, pp. 182–189. Springer, Berlin & Heidelberg (2001) · Zbl 1013.68226
[24] Järvinen J.: ’On the Structure of Rough Approximations (Extended Abstract)’. In: Alpigini, J.J., Peters, J.F., Skowron, A., Zhong, N.(eds) Rough Sets and Current Trends in Computing: Third International Conference, RSCTC 2002, Malvern, PA, USA, October 14–16, 2002. Proceedings LNAI 2475, pp. 123–130. Springer, Berlin & Heidelberg (2002)
[25] Järvinen J.: ’The Ordered Set of Rough Sets’. In: Tsumoto, S., Słowiński, R., Komorowski, J., Grzymala-Busse, J.W. (eds) Rough Sets and Current Trends in Computing: 4th International Conference, RSCTC 2004 Uppsala, Sweden, June 1–5, 2004 Proceedings, LNAI 3066, pp. 49–58. Springer, Berlin & Heidelberg (2004)
[26] Järvinen, J., ’Topologies and Lattice Structures in Rough Set Theory’, presented at Algebra and its Applications, Kokõ, May 5–7, 2006; available on-line at http://www.cs.utu.fi/jjarvine/Slides/17.pdf .
[27] Kalmbach G.: Orthomodular Lattices. Academic Press, London & New York (1983) · Zbl 0512.06011
[28] Komorowski J., Pawlak Z., Polkowski L., Skowron A.: ’Rough Sets: A Tutorial’. In: Pal, S.K., Skowron, A. (eds) Rough Fuzzy Hybridization: A New Trend in Decision-Making, pp. 3–98. Springer, Singapore (1999)
[29] Kondo, M., ’Algebraic Approach to Generalized Rough Sets’, in D. Ślezak, G. Wang, M. Szczuka, I. Düntsch and Y.Y. Yao (eds.), Rough Sets, Fuzzy Sets, Data Mining, and Granular Computing: 10th International Conference, RSFDGrC 2005, Regina, Canada, August 31 – September 3, 2005, Proceedings, Part I, LNAI 3641, Berlin & Heidelberg: Springer, 2005, pp. 132–140. · Zbl 1134.68509
[30] Koons R.C.: ’A New Solution to the Sorites Problem’. Mind 103(4), 439–449 (1994)
[31] Koopman B.O.: ’The Axioms and Algebra of Intuitive Probability’. Annals of Mathematics 41(2), 269–292 (1940) · Zbl 0024.05001
[32] Kortelainen J.: ’On relationship between modified sets, topological spaces and rough sets’. Fuzzy Sets and Systems 61(1), 91–95 (1994) · Zbl 0828.04002
[33] Lewis, D.K., ’Probabilities of Conditionals and Conditional Probabilities’, Philosophical Review 85 (3):297–315, 1976; reprinted with postscript in Lewis, Philosophical Papers, Vol. 2, Oxford: Oxford University Press, 1986, pp. 133–156.
[34] Mazurkiewicz, S., Podstawy Rachunka Prawdopodobienstwa, Warsaw: Państowe Wydawnictwo Naukawe, 1956.
[35] Milne P.: ’The Foundations of Probability and Quantum Mechanics’. Journal of Philosophical Logic 22(2), 129–68 (1993) · Zbl 0774.60001
[36] Milne P.: ’Bruno de Finetti and the Logic of Conditional Events’. British Journal for the Philosophy of Science 48(2), 195–232 (1997) · Zbl 0948.03015
[37] Milne P.: ’Algebras of Intervals and a Logic of Conditional Assertions’. Journal of Philosophical Logic 33(5), 497–548 (2004) · Zbl 1055.03008
[38] Milne, P., ’Conditional probability, conditional events, and single-case propensities’, in Petr Hájek, Luis Valdés-Villanueva and Dag Westerståhl (eds.), Logic, Methodology, and Philosophy of Science: Proceedings of Twelfth International Congress, London: King’s College Publications, 2005, pp. 315–331. · Zbl 1095.03504
[39] Orłlowska E.: ’The Semantics of Vague Concepts’. In: Dorn, G., Weingartner, P. (eds) Foundations of Logic and Linguistics., pp. 465–482. Plenum, New York (1984)
[40] Pawlak Z.: ’Some Issues on Rough Sets’. In: Peters, J.F., Skowron, A., Grzymałla-Busse, J.W., Kostek, B., Świniarski, R.W., Szczuka, M.S. (eds) Transactions on Rough Sets I, LNCS 3100., pp. 1–58. Springer, Berlin & Heidelberg (2004)
[41] Pawlak Z.: ’A Treatise on Rough Sets’. In: Peters, J.F., Skowron, A. (eds) Transactions on Rough Sets IV, LNCS 3700., pp. 1–17. Springer, Berlin & Heidelberg (2005) · Zbl 1136.68535
[42] Pawlak Z., Grzymala-Busse J., Slowinski R., Ziarkio W.: ’Rough Sets’. Communications of the ACM 38(11), 89–95 (1995)
[43] Pomykała J., Pomykała J.A.: ’The Stone Algebra of Rough Sets’. Bulletin of the Polish Academy of Sciences: Mathematics 36, 495–508 (1988) · Zbl 0786.04008
[44] Pták, P., and S. Pulmannová, Orthomodular Structures as Quantum Logics (Fundamental Theories of Physics, 44), Kluwer: Dordrecht, 1991.
[45] Read S.: Thinking About Logic: An introduction to the philosophy of logic. Oxford University Press, Oxford (1994)
[46] Rogers C.A.: Hausdorff Measures. Cambridge University Press, Cambridge (1970) · Zbl 0204.37601
[47] Slowinski R., Vanderpooten D.: ’A Generalized Definition of Rough Approximations Based on Similarity’. IEEE Transactions on Knowledge and Data Engineering 12(2), 331–336 (2000) · Zbl 05108848
[48] Walley, P., Statistical Reasoning with Imprecise Probabilities (Monographs on Statistics and Applied Probability, 42), London: Chapman & Hall, 1991. · Zbl 0732.62004
[49] Williamson T.: Vagueness. London, Routledge (1994)
[50] Yao Y.Y.: ’Relational interpretation of neighborhood operators and rough set approximation operators’. Information Sciences 111(1–4), 239–259 (1998) · Zbl 0949.68144
[51] Yao Y.Y.: ’Probabilistic approaches to rough sets’. Expert Systems 20(5), 287–297 (2003) · Zbl 05653445
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