The Qualitative Characteristics of Combining Evidence with Discounting
The qualitative characteristics of the combining evidence with the help of Dempster’s rule with discounting is studied in this paper in the framework of Dempster-Shafer theory. The discount coefficient (discounting rate) characterizes the reliability of i
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Abstract The qualitative characteristics of the combining evidence with the help of Dempster’s rule with discounting is studied in this paper in the framework of Dempster-Shafer theory. The discount coefficient (discounting rate) characterizes the reliability of information source. The conflict between evidence and change of ignorance after applying combining rule are considered in this paper as important characteristics of quality of combining. The quantity of ignorance is estimated with the help of linear imprecision index. The set of crisp and fuzzy discounting rates for which the value of ignorance after combining does not increases is described. Keywords Belief functions · Discount method · Imprecise index
1 Introduction The study of combining rules of evidence occupies an important place in the belief functions theory. A combining rule puts in correspondence to two or more evidences the one evidence. Dempster’s rule [4] was the first from combining rules. The review of some popular combining rules can be found in [10]. There is no combining rule which give a plausible aggregation of information in all cases regardless of context. The prognostic quality of combining evidence is evaluated with the help of some characteristics. The reliability of sources of information, the conflict measure of evidence [7], the degree of independence of evidence are a priori characteristics of quality of combining. The amount of change of ignorance after the use of a combining rule is the most important a posteriori characteristic [8]. The amount of ignorance contained in evidence may be estimated with the help of imprecision indices [2]. The generalized Hartley’s measure is an example of such index [5]. It is known, for example, that the amount of ignorance does not increase when used Dempster’s rule for non-conflicting evidences. Dempster’s rule can be considered as an optimistic rule in this sense [8]. On the contrary, Dubois and Prade’s disjunctive consensus A. Lepskiy (B) Higher School of Economics, 20 Myasnitskaya Ulitsa, 101000 Moscow, Russia e-mail: [email protected] © Springer International Publishing Switzerland 2017 M.B. Ferraro et al. (eds.), Soft Methods for Data Science, Advances in Intelligent Systems and Computing 456, DOI 10.1007/978-3-319-42972-4_39
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rule [6] has a pessimistic character in the sense that amount of ignorance does not decrease after applying such a rule. The discount method is one of the approaches where the reliability of information source is taken into account. This method was proposed by Shafer [11]. The discount coefficient (discounting rate) characterizes the reliability of information source. The discount method with Dempster’s rule may be pessimistic rule or optimistic rule in depending on the values of discounting rates. The generalizations of the discount method were considered in several papers. In particular, Smets [12] introduced a family of combination rules known as α-junctions. Pichon and Denoeux [9] have established the link between the parameter of α
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