On the Dempster-Shafer framework and new combination rules.

*(English)*Zbl 0629.68092The author discusses the known instabilities of Dempster composition of upper and lower probabilities in the Shafer framework. Dempster composition proceeds in two stages. In the first one, the composition of information from different sources may result in the assignment of a nonzero probability k to the empty event. To eliminate this, one divides the computed probabilities by 1-k. The author proposes instead to leave probabilities of subevents unreduced and to add k to the mass assigned to the universal set. {K, defined as \(\log(1-k)\), is a misprint on p. 116}. He shows in examples that this composition rule is better behaved, and he proves a few elementary theorems. Alternatively, he proposes to determine weights for given information that maximizes a complicated function measuring the quality of combined evidence while retailing Dempster’s rule. \(\{\) The corresponding nonlinear programming problem seems computationally untractable once the number of data is not very small\(\}\). At the end, he proposes a third scheme to combine information about the competence of an expert with information about that person’s veracity by combining Dempster composition with extension and projection operations.

{The results ascribed on p. 110 to G. Shafer, A mathematical theory of evidence (1976; Zbl 0359.62002), are due to A. P. Dempster, Upper and lower probabilities induced by a multivalued mapping [Ann. Math. Stat. 38, 325-339 (1967; Zbl 0168.175)]. From the mathematical point of view, the paper like all papers based on Shafer’s book instead of Dempster’s paper totally neglects statistical independence which is a necessary and sufficient condition for the validity of Dempster’s rule. For example, one would assume that an expert has little incentive to be untruthful if he truly competent; this would disqualify Dempster’s rule from being used in the third procedure. The mathematical background of instability in Dempster composition has been investigated by the reviewer, Probabilistic propositional logic [Polytech. Notes Artif. Intell. 4, No.4 (1987; Zbl 0626.03013)].}

{The results ascribed on p. 110 to G. Shafer, A mathematical theory of evidence (1976; Zbl 0359.62002), are due to A. P. Dempster, Upper and lower probabilities induced by a multivalued mapping [Ann. Math. Stat. 38, 325-339 (1967; Zbl 0168.175)]. From the mathematical point of view, the paper like all papers based on Shafer’s book instead of Dempster’s paper totally neglects statistical independence which is a necessary and sufficient condition for the validity of Dempster’s rule. For example, one would assume that an expert has little incentive to be untruthful if he truly competent; this would disqualify Dempster’s rule from being used in the third procedure. The mathematical background of instability in Dempster composition has been investigated by the reviewer, Probabilistic propositional logic [Polytech. Notes Artif. Intell. 4, No.4 (1987; Zbl 0626.03013)].}

Reviewer: H.Guggenheimer

##### MSC:

68T99 | Artificial intelligence |

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##### References:

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