Cautious Conjunctive Merging of Belief Functions

Abstract

Abstract. When merging belief functions, Dempster rule of combination is justified only when information sources can be considered as independent. When this is not the case, one must find out a cautious merging rule that adds a minimal amount of information to the inputs. Such a rule is said to follow the principle of minimal commitment. Some conditions it should comply with are studied. A cautious merging rule based on maximizing expected cardinality of the resulting belief function is proposed. It recovers the minimum operation when specialized to possibility distributions. This form of the minimal commitment principle is discussed, in particular its discriminating power and its justification when some conflict is present between the belief functions.

Can the Minimum Rule of Possibility Theory Be Extended to Belief Functions?

“…Also, existing results tend to show that the notion of q-inclusion is more natural when working within the TBM interpretation [14,7].…”

“…One of such criteria, that is used in other approaches to the cautious merging of belief function [14,7], is the expected cardinality of a belief structure (m, F), that we denote by C |(m,F)| and whose value is…”

“…We define a belief structure (m ∩ , F ∩ ) resulting from a conjunctive merging of (m 1 , F 1 ), (m 2 , F 2 ) as the result of the following procedure [7]: 1. A joint mass distribution m is built on V 2 , having focal sets of the form A × B where A ∈ F 1 , B ∈ F 2 and preserving m 1 , m 2 as marginals.…”

“…Considering the idempotent rule of possibility theory and this property, it makes sense to ask for inequality (7) to become an equality. We now study two different ways to formulate this requirement on conjunctively merged belief structures.…”

“…Cattaneo [2] proposes to first consider the set of merged belief functions minimizing the resulting conflict (hence maximizing the result coherence) and then to select the most cautious belief function among them. Recently [7], we have proposed to maximize expected cardinality of belief functions in order to find a cautiously merged belief function. Although different, all these approaches agree on the fact that a cautious conjunctive merging rule should satisfy the property of idempotence, as this property ensures that the same information supplied by two dependent sources will remain unchanged after merging.…”