A vector and geometry interpretation of basic probability assignment in Dempster‐Shafer theory

Abstract

Because of the superiority in dealing with uncertainty expression, Dempster-Shafer theory (D-S theory) is widely used in decision theory. In D-S theory, the basic probability assignment (BPA) is the basis and core. Recently, some researchers represent BPA on a N -dimension frame of discernment (FOD) as 2 N -dimension vector in Descartes coordinate system. This representation treats a BPA as a point in the 2 N -dimensional space. A new vector and geometry interpretation of BPA is proposed in this paper.The BPA on a N -dimension FOD is represented as N -dimension vector with parameters in this method.Then BPA is expressed as subset of N -dimension Cartesian space rather than a point. The proposed method is a new way to represent BPA with vector and geometry. The essence of this method is to convert BPA to probability distribution with parameters.The applications of this representation method in D-S theory have been studied. Based on this method, problems in D-S theory can be solved, which include the fusion of BPAs, the distance between BPAs, the correspondence between BPA and probability, and the entropy of BPAs.basic probability assignment, data fusion, dempster-shafer theory, distance, entropy Dempster-Shafer theory (D-S theory), 1,2 as an important and widely used uncertain reasoning method, has been receiving increasingly attention. 3,4 D-S theory assigns probabilities to the power set of events, so it can effectively deal with uncertainty and unknown problems. Because of the effectiveness in pattern recognition, 5,6 decision making, 7,8,9,10,11,12 reasoning, 13,14,15 and uncertainty analysis, 16,17,18,19 D-S theory has been applied in various fields. 20,21,22,23 In D-S theory, A complete set of incompatible basic hypotheses is called a frame of discernment (FOD), which represents all possible answers to a problem. The degree of trust assigned to each subset of FOD is called the basic probability assignment (BPA). Since the N -dimension FOD has 2 N subsets, some researchers represent the BPA as 2 N -dimension vector in Descartes coordinate system. 24,25 A BPA is a point in the 2 N -dimensional space in this representation method.Based on the idea of converting BPA to probability distribution (PD), a new interpretation of D-S theory is proposed in this paper. The BPA on a N -dimension FOD is represented as N -dimension vector with parameters in this method. Each dimension indicates a hypothesis in FOD. In fact, BPA is represented as a subset of N -dimension space rather than a point because of the variable parameters. In this way, BPAs have an intuitive vector and geometry representation. Based on this representation method, several problems in D-S theory have been studied, which include the fusion of BPAs, the distance between BPAs, the correspondence between BPA and probability, and the entropy of BPAs. The example shows that our method provides a new way to solve these basic problems of D-S theory, and can get reasonable results.Combination of BPAs from different information sources is a major problem in D...

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