On the justification of Dempster's rule of combination

Abstract

In Dempster-Shafer theory it is claimed that the pooling of evidence is reflected by Dempster's rule of combination, provided certain requirements are met. The justification of this claim is problematic, since the existing formulations of the requirements for the use of Dempster's rule are not completely clear. In this paper, randomly coded messages, Shafer's canonical examples for Dempster-Shafer theory, are employed to clarify these requirements and to evaluate Dempster's rule. The range of applicability of Dempster-Shafer theory will turn out to be rather limited. Further, it will be argued that the mentioned requirements do not guarantee the validity of the rule and some possible additional conditions will be described. Acknowledgment: I would like to thank Gerard Renardel de Lavalette for drawing my attention to the problem of justifying Dernpster's rule of combination.

Reinvestigating Dempster's Idea on Evidence Combination

“…The condition of using the rule is stated as "two or more pieces of evidence are based on distinct bodies of evidence" [14]. This description is a bit confusing and causes a lot of misapplications and counterintuitive results [22].…”

“…This requirement is called DS-independent in [22]. Now the two approaches in Dempster's combination framework can be summarised as follows.…”

“…The idea of describing and judging dependent relations among the original probability spaces has also been mentioned implicitly by Shafer [15], Shafer and Tversky [19] and Voorbraak [22] but not explicitly defined at the original information level.…”

“…The first two examples come from [16] where the former shows that when two pieces of evidence are statistically independent, both approaches in Dempster's combination framework are applicable, and the latter shows that when they are not statistically independent, only the first approach works. The third example ( [22]) not only demonstrates that the careless application of Dempster's combination rule will yield a wrong result but also reveals the possibility of using Approach 1 to combine two dependent pieces of evidence when their common probability space is known (as also investigated in [16], [17], [12]), although Dempster's original intention was aimed at dealing with independent probability spaces.…”

“…Various discussions and criticisms of the rule have appeared in the literature. A mathematical description of the condition of applying Dempster's combination rule is formalised in [22].…”