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Williams coherence and beyond. (English) Zbl 1214.68403
Summary: We discuss the consistency concept of Williams coherence for imprecise conditional previsions, presenting a variant of this notion, which we call W-coherence. It is shown that W-coherence ensures important consistency properties and is quite general and well-grounded. This is done comparing it with alternative or anyway similar known and less known consistency definitions. The common root of these concepts is that they variously extend to imprecision the subjective probability approach championed by de Finetti. The analysis in the paper is also helpful in better clarifying several little investigated aspects of these notions.

68T37 Reasoning under uncertainty in the context of artificial intelligence
60A05 Axioms; other general questions in probability
60A99 Foundations of probability theory
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[1] Armstrong, T.E.; Sudderth, W.D., Locally coherent rates of exchange, Annals of statistics, 17, 1394-1408, (1989) · Zbl 0743.60006
[2] Artzner, P.; Delbaen, F.; Eber, S.; Heath, D., Coherent measures of risk, Mathematical finance, 9, 203-228, (1999) · Zbl 0980.91042
[3] L. Crisma, Events and conditional events: a subjectivistic approach, in: The notion of event in probabilistic epistemology, Pubbl. n. 2, Dip. Mat. Appl. ‘B. de Finetti’, Ediz. Lint, Trieste, 1996, pp. 43-89.
[4] L. Crisma. Introduzione alla teoria delle probabilità coerenti. Ediz. EUT, Trieste, available online at <http://www.openstarts.units.it/dspace/items-by-author?author=Crisma%2C+Lucio>, 2006.
[5] G. de Cooman, F. Hermans, On coherent immediate prediction: connecting two theories of imprecise probabilities, in: G. de Cooman, J. Vejnarová, M. Zaffalon (Eds.), Proc. ISIPTA’07, Prague, Czech Republic, 2007, pp. 97-106.
[6] de Finetti, B., Sulla proprietà conglomerativa delle probabilità subordinate, Rendiconti R. ist. lombardo di scienze e lettere, 43, 6-10, 339-343, (1930) · JFM 56.0445.02
[7] B. de Finetti, Sull’impostazione assiomatica del calcolo delle probabilità, Annali triestini dell’Università di Trieste, XIX 2, 1949, pp. 29-81.
[8] de Finetti, B., Theory of probability, vol. 1, (1974), Wiley
[9] Holzer, S., On coherence and conditional prevision, Bollettino dell’unione matematica italiana, 4, 1, 441-460, (1985) · Zbl 0584.60001
[10] S. Maaß, Continuous linear representations of coherent lower previsions, in: J.M. Bernard, T. Seidenfeld, M. Zaffalon (Eds.), Proc. ISIPTA’03, Lugano (CH), 2003, pp. 372-382.
[11] E. Miranda, G. de Cooman, Coherence and independence for non-linear spaces, Research Report TR05/10, Universidad Rey Juan Carlos, available at <http://bayes.escet.urjc.es/publicaciones/wp05-10.pdf>, 2005.
[12] E. Miranda, M. Zaffalon, Coherence graphs. Artificial Intelligence, in press, doi:10.1016/j.artint.2008.09.001. · Zbl 1181.03023
[13] Pelessoni, R.; Vicig, P., Convex imprecise previsions, Reliable computing, 9, 6, 465-485, (2003) · Zbl 1037.60003
[14] Pelessoni, R.; Vicig, P., Uncertainty modelling and conditioning with convex imprecise previsions, International journal of approximate reasoning, 39, 2-3, 297-319, (2005) · Zbl 1123.62077
[15] R. Pelessoni, P. Vicig, Envelope theorems and dilation with convex conditional previsions, in: F.G. Cozman, R. Nau, T. Seidenfeld (Eds.), Proc. ISIPTA’05, Pittsburgh, PA, 2005, pp. 266-275. · Zbl 1123.62077
[16] R. Pelessoni, P. Vicig. Some bounds for conditional lower previsions, in: G. de Cooman, J. Vejnarová, M. Zaffalon (Eds.), Proc. ISIPTA’07, Prague, Czech Republic, 2007, pp. 337-346. · Zbl 1211.62012
[17] Regazzini, E., Finitely additive conditional probability, Rend. sem. mat. fis., 55, 69-89, (1985) · Zbl 0631.60001
[18] Schervish, M.J.; Seidenfeld, T.; Kadane, J.B., The extent of non-conglomerability of finitely additive probabilities, Zeitschrift für wahrscheinlichkeitstheorie, 66, 205-226, (1984) · Zbl 0525.60003
[19] Schmeidler, D., Cores of exact games, I, Journal of mathematical analysis and applications, 40, 214-225, (1972) · Zbl 0243.90071
[20] Shafer, G.; Vovk, V., Probability and finance, (2001), Wiley-Interscience · Zbl 0985.91024
[21] Shafer, G.; Gillett, P.R.; Scherl, R.B., A new understanding of subjective probability and its generalization to lower and upper previsions, International journal of approximate reasoning, 33, 1, 1-49, (2003) · Zbl 1092.68098
[22] Troffaes, M.C.M.; de Cooman, G., Lower previsions for unbounded random variables, (), 146-155
[23] Troffaes, M.C.M., Conditional lower previsions for unbounded random quantities, (), 201-209 · Zbl 1116.68101
[24] Vicig, P., Epistemic independence for imprecise probabilities, International journal of approximate reasoning, 24, 235-250, (2000) · Zbl 0995.68121
[25] Vicig, P.; Zaffalon, M.; Cozman, F.G., Notes on ‘notes on conditional previsions’, International journal of approximate reasoning, 44, 358-365, (2007) · Zbl 1114.60004
[26] Walley, P., Statistical reasoning with imprecise probabilities, (1991), Chapman and Hall · Zbl 0732.62004
[27] Walley, P.; Pelessoni, R.; Vicig, P., Direct algorithms for checking consistency and making inferences from conditional probability assessments, Journal of statistical planning and inference, 126, 1, 119-151, (2004) · Zbl 1075.62002
[28] K. Weichselberger, T. Augustin, On the symbiosis of two concepts of conditional interval probability, in: J.M. Bernard, T. Seidenfeld, M. Zaffalon (Eds.), Proc. ISIPTA’03, Lugano (CH), 2003, pp. 608-629.
[29] Williams, P.M., Notes on conditional previsions, International journal of approximate reasoning, 44, 366-383, (2007), revised version of: notes on conditional previsions. Research Report, School of Math. and Phys. Science, University of Sussex, 1975. · Zbl 1114.60005
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