×

zbMATH — the first resource for mathematics

Belief function independence: I. The marginal case. (English) Zbl 1015.68207
Summary: We study the notion of marginal independence between two sets of variables when uncertainty is expressed by belief functions as understood in the context of the transferable belief model. We define the concepts of non-interactivity and irrelevance, that are not equivalent. Doxastic independence for belief functions is defined as irrelevance and irrelevance preservation under Dempster’s rule of combination. We prove that doxastic independence and non-interactivity are equivalent.

MSC:
68T37 Reasoning under uncertainty in the context of artificial intelligence
PDF BibTeX XML Cite
Full Text: DOI Link
References:
[1] Benferhat, S.; Dubois, D.; Prade, H., Expressing independence in possibilistic framework and its application to default reasoning, (), 150-153
[2] Ben Yaghlane, B.; Smets, Ph.; Mellouli, K., Independence concepts for belief functions, (), 357-364
[3] Campos, L.M.; Huete, J.F., Independence concepts in upper and lower probabilities, (), 85-96
[4] Campos, L.M.; Huete, J.F.; Moral, S., Possibilistic independence, (), 69-73
[5] Couso, I.; Moral, S.; Walley, P., Examples of independence for imprecise probabilities, (), 121-130
[6] Cozman, F.G., Irrelevance and independence axioms in quasi-Bayesian theory, (), 128-136 · Zbl 0946.62004
[7] Dawid, A.P., Conditional independence in statistical theory, Journal of the royal statistical society series B, 41, 1-31, (1979) · Zbl 0408.62004
[8] Dawid, A.P., Conditional independence, Encyclopedia of statistical science (update), vol. 3, (1999), Wiley New York · Zbl 0408.62004
[9] Fonck, P., Conditional independence in possibility theory, Uncertainty in artificial intelligence, 221-226, (1994)
[10] Geiger, D.; Verma, T.; Pearl, J.T., Identifying independence in Bayesian networks, Networks, 20, 507-533, (1990) · Zbl 0724.05066
[11] Hisdal, E., Conditional possibilities independence and non interaction, Fuzzy sets and systems, 1, (1978) · Zbl 0393.94050
[12] Klawonn, F.; Smets, P., The dynamic of belief in the transferable belief model and specialization – generalization matrices, (), 130-137
[13] Kohlas, J.; Monney, P.A., A mathematical theory of hints: an approach to dempster – shafer theory of evidence, Lecture notes in economics and mathematical systems, vol. 425, (1995), Springer Berlin · Zbl 0833.62005
[14] C.T.A. Kong, A Belief Function Generalization of Gibbs Ensemble, Joint Technical Report, S-122 Harvard University and N239 University of Chicago, Departments of Statistics, 1988
[15] Lauritzen, S.L.; Dawid, A.P.; Larsen, B.N.; Leimer, H.G., Independence properties of directed Markov fields, Networks, 20, 5, 491-505, (1990) · Zbl 0743.05065
[16] Pearl, J., Probabilistic reasoning in intelligent systems: networks of plausible inference, (1988), Morgan Kaufmann Los Altos, CA
[17] Shafer, G., Mathematical theory of evidence, (1976), Princeton University Press Princeton · Zbl 0359.62002
[18] Shenoy, P.P., Conditional independence in valuation-based systems, International journal of approximate reasoning, 10, 203-234, (1994) · Zbl 0821.68114
[19] Smets, P., The transferable belief model for quantified belief representation, (), 267-301 · Zbl 0939.68112
[20] Smets, P.; Kennes, R., The transferable belief model, Artificial intelligence, 66, 191-234, (1994) · Zbl 0807.68087
[21] Spohn, W., Ordinal conditional functions: a dynamic theory of epistemic states, (), 105-134
[22] Studeny, M., Formal properties of conditional independence in different calculi of artificial intelligence, (), 341-348
[23] Vejnarova, J., Conditional independence relations in possibility theory, (), 343-351
[24] Vicig, P., Epistemic independence for imprecise probabilities, International journal of approximate reasoning, 24, 235-250, (2000) · Zbl 0995.68121
[25] Walley, P., Statistical reasoning with imprecise probabilities, (1991), Chapman and Hall London · Zbl 0732.62004
[26] Xu, H., An efficient implementation of the belief function propagation, (), 425-432
[27] Zadeh, L.A., Fuzzy sets as a basis for 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.