Evidential Matrix Metrics as Distances Between Meta-Data Dependent Bodies of Evidence

Abstract

As part of the theory of belief functions, we address the problem of appraising the similarity between bodies of evidence in a relevant way using metrics. Such metrics are called evidential distances and must be computed from mathematical objects depicting the information inside bodies of evidence. Specialization matrices are such objects and, therefore, an evidential distance can be obtained by computing the norm of the difference of these matrices. Any matrix norm can be thus used to define a full metric. In this paper, we show that other matrices can be used to obtain new evidential distances. These are the α -specialization and α -generalization matrices and are closely related to the α -junctive combination rules. We prove that any L(1) norm-based distance thus defined is consistent with its corresponding α -junction. If α > 0 , these distances have in addition relevant variations induced by the poset structure of the belief function domain. Furthermore, α -junctions are meta-data dependent combination rules. The meta-data involved in α -junctions deals with the truthfulness of information sources. Consequently, the behavior of such evidential distances is analyzed in situations involving uncertain or partial meta-knowledge about information source truthfulness.

Complementary Lipschitz continuity results for the distribution of intersections or unions of independent random sets in finite discrete spaces

“…See D for proof. As compared to previous Lipschitz continuity results [25,26], specialization distances have a greater time complexity as compared to commonality ones. Indeed, although the construction of specialization distances can be sped up [24], the time complexity for the specialization distance is quadratic in N. More precisely, the time complexity to build a specialization matrix is…”

“…The α-junctions allow to combine mass functions in several situations ranging between these two extreme cases. This interpretation is documented in [34,33,26,21].…”

“…In [26], two families of mass function distances are introduced. Each of them relies on a given type of evidential matrix and a matrix norm.…”

“…Another bijective correspondence also exists between a given mass function m i and an α-generalization matrix G (α) i . The main result in [26] states that, for any α ∈ [0; 1], the distance induced by the L 1 matrix norm of the difference between a pair of α-specialization matrices is consistent with the α-conjunctive rule. Likewise, for any α ∈ [0; 1], the distance induced by the L 1 matrix norm of the difference between a pair of α-generalization matrices is consistent with the α-disjunctive rule.…”

“…The authors proved that the consistency under study holds between some belief function distances and combination operators that yield the distribution of the intersection or union of independent random sets. The results that we introduce in this paper involves distances that are computationally more tractable than those introduced in [25,26] but also rely on independence assumptions. In the random set literature, many results regarding unions of i.i.d.…”