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Dempster–Shafer theory
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==Formal definition== Let ''X'' be the ''[[Universe (mathematics)|universe]]'': the set representing all possible states of a system under consideration. The [[power set]] :<math>2^X \,\!</math> is the set of all subsets of ''X'', including the [[empty set]] <math>\emptyset</math>. For example, if: :<math>X = \left \{ a, b \right \} \,\!</math> then :<math>2^X = \left \{ \emptyset, \left \{ a \right \}, \left \{ b \right \}, X \right \}. \,</math> The elements of the power set can be taken to represent propositions concerning the actual state of the system, by containing all and only the states in which the proposition is true. The theory of evidence assigns a belief mass to each element of the power set. Formally, a function :<math>m: 2^X \rightarrow [0,1] \,\!</math> is called a ''basic belief assignment'' (BBA), when it has two properties. First, the mass of the empty set is zero: :<math>m(\emptyset) = 0. \,\!</math> Second, the masses of all the members of the power set add up to a total of 1: :<math>\sum_{A \in 2^X} m(A) = 1.</math> The mass ''m''(''A'') of ''A'', a given member of the power set, expresses the proportion of all relevant and available evidence that supports the claim that the actual state belongs to ''A'' but to no particular subset of ''A''. The value of ''m''(''A'') pertains ''only'' to the set ''A'' and makes no additional claims about any subsets of ''A'', each of which have, by definition, their own mass. From the mass assignments, the upper and lower bounds of a probability interval can be defined. This interval contains the precise probability of a set of interest (in the classical sense), and is bounded by two non-additive continuous measures called '''belief''' (or '''support''') and '''plausibility''': :<math>\operatorname{bel}(A) \le P(A) \le \operatorname{pl}(A).</math> The belief bel(''A'') for a set ''A'' is defined as the sum of all the masses of subsets of the set of interest: :<math>\operatorname{bel}(A) = \sum_{B \mid B \subseteq A} m(B). \, </math> The plausibility pl(''A'') is the sum of all the masses of the sets ''B'' that intersect the set of interest ''A'': :<math>\operatorname{pl}(A) = \sum_{B \mid B \cap A \ne \emptyset} m(B). \, </math> The two measures are related to each other as follows: :<math>\operatorname{pl}(A) = 1 - \operatorname{bel}(\overline{A}).\,</math> And conversely, for finite ''A'', given the belief measure bel(''B'') for all subsets ''B'' of ''A'', we can find the masses ''m''(''A'') with the following inverse function: :<math>m(A) = \sum_{B \mid B \subseteq A} (-1)^{|A-B|}\operatorname{bel}(B) \, </math> where |''A'' − ''B''| is the difference of the cardinalities of the two sets.<ref name=Sentz-Ferson/> It [[Logical consequence|follows from]] the last two equations that, for a finite set ''X'', one needs to know only one of the three (mass, belief, or plausibility) to deduce the other two; though one may need to know the values for many sets in order to calculate one of the other values for a particular set. In the case of an infinite ''X'', there can be well-defined belief and plausibility functions but no well-defined mass function.<ref>[[Joseph Halpern|J.Y. Halpern]] (2017) ''[https://books.google.com/books?id=ih-QDgAAQBAJ&q=%22Dempster%E2%80%93Shafer%22 Reasoning about Uncertainty]'' MIT Press</ref>
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