×

zbMATH — the first resource for mathematics

Possibility theory: conditional independence. (English) Zbl 1092.68094
Summary: The subtle notion of conditioning is controversial in several contexts, for example in possibility theory where, in fact, different definitions have been introduced. We refer to a general axiomatic definition of conditional possibility and then we deal with “partial assessments” on (not necessarily structured) domains containing only elements of interest. We study a notion of coherence, which assures the extendability of an assessment as a conditional possibility and we introduce a procedure for checking coherence. Moreover, we propose a definition of independence for conditional possibility, which avoids some counterintuitive situations, and we study its main properties in order to compare it with other definitions introduced in literature. Then, we check which properties among the graphoid ones are satisfied: this allows to compare our definition with other independence notions given in the context of other uncertainty formalisms. This analysis is relevant for graphical models in order to single out and visualize dependence relations among random variables.

MSC:
68T37 Reasoning under uncertainty in the context of artificial intelligence
94D05 Fuzzy sets and logic (in connection with information, communication, or circuits theory)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Amor, N.B.; Benferhat, S., Graphoid properties of qualitative possibilistic independence relations, Internat. J. uncertainty, fuzziness knowledge-based systems, 13, 1, 59-96, (2005) · Zbl 1096.68147
[2] Amor, N.B.; Mellouli, K.; Benferhat, S.; Dubois, D.; Prade, H., A theoretical framework for possibilistic independence in a weakly ordered setting, Internat. J. uncertainty, fuzziness knowledge-based systems, 10, 2, 117-155, (2002) · Zbl 1084.68126
[3] S. Benferhat, D. Dubois, H. Prade, Expressing independence in a possibilistic framework and its application to default reasoning, in: Cohn, (Ed.), 11th European Conf. in Artificial Intelligence. Wiley, New York, 1994, pp. 150-154.
[4] Bouchon-Meunier, B.; Coletti, G.; Marsala, C., Independence and possibilistic conditioning, Ann. math. artif. intell., 35, 107-124, (2002) · Zbl 1004.60001
[5] Bouchon-Meunier, B.; Coletti, G.; Marsala, C., Conditional possibility and necessity, (), 59-71 · Zbl 1015.68191
[6] Coletti, G.; Scozzafava, R., Zero probabilities in stochastical independence, (), 185-196
[7] Coletti, G.; Scozzafava, R., Stochastic independence in a coherent setting, Ann. math. artif. intell., 35, 151-176, (2002) · Zbl 1005.60007
[8] G. Coletti, R. Scozzafava, Probabilistic Logic in a Coherent Setting, Trends in Logic, vol. 15, Kluwer, Dordrecht, Boston, London, 2002. · Zbl 1005.60007
[9] G. Coletti, B. Vantaggi, Independence in Conditional Possibility Theory, Proc. of 10th Internat. Conf. IPMU 2004, Italy, pp. 849-856. · Zbl 1092.68094
[10] A.P. Dawid, Conditional independence in statistical theory, J. Roy. Statist. Soc. B (1979) 15-31. · Zbl 0408.62004
[11] de Campos, L.M.; Huete, J.F., Independence concepts in possibility theory: part I, Fuzzy sets and systems, 103, 127-152, (1999) · Zbl 0951.68150
[12] de Cooman, G., Possibility theory II: conditional possibility, Internat. J. general systems, 25, 325-351, (1997) · Zbl 0955.28013
[13] de Cooman, G., Possibility theory III: possibilistic independence, Internat. J. general systems, 25, 353-371, (1997) · Zbl 0955.28014
[14] de Dombal, F.T.; Gremy, F., Decision making and medical care, (1976), North-Holland Amsterdam
[15] de Finetti, B., Sull’impostazione assiomatica del calcolo delle probabilità, Annali univ. di trieste, 19, 29-81, (1949), (English translation in: Probability, Induction, Statistics, 1972, Wiley, London, (Chapter 5)) · Zbl 0036.20703
[16] D. Dubois, L. Fariñas del Cerro, A. Herzig, H. Prade, An ordinal view of independence with application to plausible reasoning, Proc. of the 10th Conf. on Uncertainty in Artificial Intelligence, vol. 1 (R. Lopez de Mantaras, D. Poole, eds.), Seattle, WA, July 29-31, 1994, pp. 195-203.
[17] D. Dubois, L. Fariñas del Cerro, A. Herzig, H. Prade, Qualitative relevance and independence: a roadmap, Proc. 15th International Joint Conference on Artificial Intelligence, Nagoya, pp. 62-67. Extended version: A roadmap of qualitative independence. In: Fuzzy Sets, Logics and Reasoning about Knowledge (Dubois, D., Prade, H., Klement, E.P., eds.), Kluwer Academic Publ., 1999, pp. 325-350. · Zbl 0943.03013
[18] Dubois, D.; Prade, H., Possibility theory, (1988), Plenum Press New York · Zbl 0645.68108
[19] L. Ferracuti, B. Vantaggi, Independence and conditional possibilities for strictly monotone triangular norms, Internat. J. Intelligent Systems 21 (2006) 299-323. · Zbl 1088.60003
[20] P. Fonck, Conditional independence in possibility theory, in: Lopez de Mantaras, Poole, (Eds.), Proc. 10th Conf. on Uncertainty in Artificial Intelligence, Morgan and Kaufmann, San Matteo, 1994, pp. 221-226.
[21] Hill, J.R., Comment on graphical models, Statist. sci., 8, 258-261, (1993)
[22] Hisdal E. Conditional possibilities independence and noninteraction, Fuzzy Sets and Systems 1 (1978) 283-297. · Zbl 0393.94050
[23] Lauritzen, S.L., Graphical models, (1996), Clarendon Press Oxford · Zbl 0907.62001
[24] Pearl, J., Probabilistic reasoning in intelligent systems: networks of plausible inference, (1988), Morgan Kaufmann Los Altos, CA
[25] Popper, K.R., The logic of scientific discovery, (1959), Routledge London · Zbl 0083.24104
[26] Rényi, A., On conditional probability spaces generated by a dimensionally ordered set of measures, Theory of probab. appl., 1, 61-71, (1956) · Zbl 0073.12302
[27] W. Spohn, On the Properties of Conditional Independence, in: Humphreys, Suppes (Eds.), Scientific Philosopher 1: Probability and Probabilistic Causality, Kluwer, Dordrecht, pp. 173-194.
[28] Studeny, M., Formal properties of conditional independence in different calculi of AI, (), 341-348
[29] Studeny, M.; Bouckaert, R.R., On chain graph models for description of conditional independence structures, Ann. statist., 26, 4, 1434-1495, (1998) · Zbl 0930.62066
[30] Vantaggi, B., Conditional independence in a coherent finite setting, Ann. math. artif. intell., 32, 287-314, (2001) · Zbl 1314.60012
[31] Vantaggi, B., The L-separation criterion for description of cs-independence models, Internat. J. approx. reasoning, 29, 291-316, (2002) · Zbl 1014.62002
[32] Vantaggi, B., Conditional independence structures and graphical models, Internat. J. uncertainty, fuzziness knowledge-based systems, 11, 5, 545-571, (2003) · Zbl 1072.68101
[33] Vejnarová, J., Conditional independence relations in possibility theory, International J. uncertainty, fuzziness knowledge-based systems, 8, 253-269, (2000) · Zbl 1113.68536
[34] Zadeh, L.A., Fuzzy sets as a basis for a theory of possibility, Fuzzy sets and systems, 1, 3-28, (1978) · Zbl 0377.04002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.