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Belief function independence. II: The conditional case. (English) Zbl 1052.68126
Summary: In Part I [ibid. 29, 47–70 (2002; Zbl 1015.68207)], we have emphasized the distinction between non-interactivity and doxastic independence in the context of the transferable belief model. The first corresponds to decomposition of the belief function, whereas the second is defined as irrelevance preserved under Dempster’s rule of combination. We had shown that the two concepts are equivalent in the marginal case. We proceed here with the conditional case. We show how the definitions generalize themselves, and that we still have the equivalence between conditional non-interactivity and conditional doxastic independence.

68T37 Reasoning under uncertainty in the context of artificial intelligence
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[1] Almond, R.G., Graphical belief modeling, (1995), Chapman & Hall London
[2] Ben Yaghlane, B.; Smets, P.; Mellouli, K., On conditional belief function independence, () · Zbl 1005.68553
[3] Ben Yaghlane, B.; Smets, P.; Mellouli, K., Independence concepts for belief functions, (), 45-58, A short version has been appeared in IPMU’2000, Information Processing and Management of Uncertainty in Knowledge-based Systems, Madrid, Vol. I, pp. 357-364, 2000 · Zbl 1015.68190
[4] Ben Yaghlane, B.; Smets, P.; Mellouli, K., Belief function independence: I. the marginal case, Int. J. approx. reasoning, 29/1, 1, 47-70, (2002) · Zbl 1015.68207
[5] Cozman, F.G., Irrelevance and independence axioms in quasi-Bayesian theory, (), 128-136 · Zbl 0946.62004
[6] Dawid, A.P., Conditional independence in statistical theory, J. roy. statist. soc., ser. B, 41, 1-31, (1979) · Zbl 0408.62004
[7] Dawid, A.P., Conditional independence, Encycl. statist. sci., 2, 146-155, (1998)
[8] Dawid, A.P., Separoids: A mathematical framework for conditional independence and irrelevance, Ann. math. artif. intell., 32, 335-372, (2001) · Zbl 1314.68308
[9] Dawid, A.P.; Studeny, M., Conditional products: an alternative approach to conditional independence, (), 32-40
[10] de Campos, L.M.; Huete, J.F.; Moral, S., Possibilistic independence, (), 69-73
[11] Dempster, A.P., Upper and lower probabilities induced by a multiple valued mapping, Ann. math. statist., 38, 325-339, (1967) · Zbl 0168.17501
[12] Fagin, R., Multivalued dependencies and a new form for relational databases, ACM trans. database syst., 2, 3, 262-278, (1977)
[13] Fonck, P., Conditional independence in possibility theory, (), 221-226
[14] Hunter, D., Graphoids and natural conditional functions, Int. J. approx. reasoning, 5, 489-504, (1991) · Zbl 0741.68088
[15] Jaffray, J.Y., Linear utility theory for belief functions, Oper. res. lett., 8, 107-112, (1989) · Zbl 0673.90010
[16] Kohlas, J.; Monney, P.A., A mathematical theory of hints: an approach to dempster – shafer theory of evidence, () · Zbl 0833.62005
[17] C.T.A. Kong, A belief function generalization of gibbs ensemble. Technical report, S-122 Harvard University and N239, 1988, University of Chicago, Department of Statistics
[18] Pearl, J., Probabilistic reasoning in intelligent systems: networks of plausible inference, (1988), Morgan Kaufmann San Mateo, CA, USA
[19] Pearl, J.; Paz, A., Graphoids: A graph based logic for reasoning about relevance relations, (), 357-363
[20] Shafer, G., A mathematical theory of evidence, (1976), Princeton University Press Princeton, NJ · Zbl 0359.62002
[21] Shafer, G.; Shenoy, P.P.; Mellouli, K., Propagating belief functions in qualitative Markov trees, Int. J. approx. reasoning, 1, 349-400, (1987) · Zbl 0641.68158
[22] Shenoy, P.P., Conditional independence in valuation-based systems, Int. J. approx. reasoning, 10, 203-234, (1994) · Zbl 0821.68114
[23] Smets, P., Belief functions: the disjunctive rule of combination and the generalized Bayesian theorem, Int. J. approx. reasoning, 9, 1-35, (1993) · Zbl 0796.68177
[24] Smets, P., Probability, possibility, belief: which for what?, (), 20-40
[25] Smets, P., The transferable belief model for quantified belief representation, (), 267-301 · Zbl 0939.68112
[26] Smets, P.; Kennes, R., The transferable belief model, Artificial intelligence, 66, 191-234, (1994) · Zbl 0807.68087
[27] Spohn, W., Ordinal conditional functions: A dynamic theory of episemic states, (), 105-134
[28] Studeny, M., Formal properties of conditional independence in different calculi of artificial intelligence, (), 341-348
[29] Vejnarova, J., Conditional independence relations in possibility theory, (), 343-351
[30] Walley, P., Statistical reasoning with imprecise probabilities, (1991), Chapman & Hall London · Zbl 0732.62004
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