Computer programming and formal systems by P. Braffort

By P. Braffort

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On the purely logical side, we should expect that any statement from a reliable source is indeed true. This allows us to write reli → ϕi to connect the auxiliary variable RELi with ϕi . With Φ + = {rel1 → ϕ1 , . . , reln → ϕn } we denote the set of all such material implications, from which we obtain a probabilistic argumentation system A + = (V ∪ W, LV ∪W , Φ + ,W, P) with W = {REL1 , . . , RELn } and P as defined above. This allows us then to compute the degrees of support and possibility for the conclusion ψ and to use them as lower and upper bounds for the target interval Y .

If we assume the least restrictive interval Xi = [0, 1] to represent a totally incompetent source, and similarly the most restrictive interval Xi = [xi , xi ] to represent a totally competent source, then ui − i surely represents the source’s degree of incompetence, from which we obtain P(compi ) = 1 − (ui − i ) = 1 − ui + i for the marginal probability of compi . Following a similar line of reasoning, we first obtain P(compi ∧ honi ) = i for the combined event compi ∧ honi of a reliable source, which then leads to P(honi ) = i P(compi ) = i 1 − u i + li for the marginal probability of honi .

Note that the independence assumption, on which Dempster’s rule is based, has raised quite some criticism with regard to the appropriateness of the rule and the theory as a whole (Zadeh, 1979). In probabilistic argumentation, these criticisms are circumvented by not explicitly formulating Dempster’s rule and thus by not giving it such a fundamental role. Another major difference is the fact that the notions of belief and plausibility in the Dempster-Shafer theory are often entirely detached from a probabilistic interpretation (for example in Smets’ Transferable Belief Model (Smets and Kennes, 1994)), whereas degrees of support and possibility are probabilities by definition.

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