A New Justification of the Unnormalized Dempster's Rule of Combination from the Least Commitment Principle

The conjunctive weight function is an equivalent representation of a non dogmatic belief function. Denœux recently proposed new rules of combination for belief functions based on pointwise combination of conjunctive weights. This paper characterizes the rules of combination based on the conjunctive weight function that have the vacuous belief function as neutral element. The main result is that the unnormalized Dempster’s rule is the least committed rule amongst those rules, for a particular informational ordering. A counterpart to this result is also presented for the disjunctive rule. Introduction The Transferable Belief Model (TBM) (Smets & Kennes 1994; Smets 1998) is a model for quantifying beliefs using belief functions (Shafer 1976). An essential mechanism of the TBM is the unnormalized Dempster’s rule of combination. This rule, referred to as the TBM conjunctive rule in this paper, allows the fusion of belief functions. Dempster’s rule and the TBM conjunctive rule have been justified by several authors. In particular, Dubois and Prade (1986a) proved the unicity of Dempster’s rule under an independence assumption. Klawonn and Smets (1992) took another path and justified the TBM conjunctive rule as being the only combination that results from an associative, commutative and least committed specialization. A limitation, which applies to both rules, is the requirement that the items of evidence combined be distinct, or in other words, that the information sources be independent. Recently, Denœux (2008) proposed a rule, called the cautious rule of combination, which does not rely on the distinctness assumption. The term cautious is reminiscent of the derivation of the rule, which is based on the least commitment principle (LCP) (Smets 1993). The LCP stipulates that one should never give more beliefs than justified by the available information, hence it promotes a cautious attitude. The cautious rule is based on the conjunctive weight function, an equivalent representation of a non dogmatic belief function. The TBM conjunctive rule can also be expressed using the conjunctive weight function, which makes it interesting to study rules based on this function. Copyright c © 2008, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. There are important differences between the TBM conjunctive rule and the cautious rule: the cautious rule is idempotent but does not have a neutral element, whereas the TBM conjunctive rule has a neutral element, the vacuous belief function, but is not idempotent. The lack of a neutral element for the cautious rule can be somewhat disturbing, hence the question: does there exist a rule based on the conjunctive weight function, which is more “cautious” than the TBM conjunctive rule and which admits the vacuous belief function as neutral element? This paper shows that the answer is no, which can be seen as a new justification of the unnormalized Dempster’s rule as the rule thus respects a central principle of the TBM. Denœux (2008) further showed that the cautious rule belongs to an infinite family of combination rules. Besides, the cautious rule is the least committed rule in this family. Interestingly, this paper shows that a similar property holds for the TBM conjunctive rule: it belongs to an infinite family of rules that admits the vacuous belief function as neutral element and it is the least committed rule in this family. The fundamental difference between those families is the existence of a neutral element. The rest of this paper is organized as follows. Necessary notions, such as the canonical decomposition of a belief function and the LCP, are first recalled in Section 2. Section 3 reviews existing rules of combination based on pointwise combination of conjunctive weights. The main result of this paper is given in Section 4. Section 5 presents results corresponding to the previous ones for rules based on the disjunctive weight function. Section 6 concludes the paper. Fundamental Concepts of the TBM Basic Definitions and Notations In this paper, the TBM (Smets & Kennes 1994; Smets 1998) is accepted as a model to quantify uncertainties based on belief functions (Shafer 1976). The beliefs held by an agent Ag on a finite frame of discernment Ω = {ω1, ..., ωK} are represented by a basic belief assignment (BBA) m defined as a mapping from 2 to [0, 1] verifying ∑ A⊆Ωm (A) = 1. Subsets A of Ω such that m(A) > 0 are called focal sets (FS) ofm. A BBAm is said to be: • normal if ∅ is not a focal set; • subnormal if ∅ is a focal set; • vacuous if Ω is the only focal set, this BBA is notedmΩ; • dogmatic if Ω is not a focal set; • categorical if it has only one focal set; • simple if if has at most two focal sets and, if it has two, Ω is one of those. A simple BBA (SBBA) m such that m (A) = 1 − w for someA 6= Ω andm (Ω) = w can be notedA. The vacuous BBA can thus be noted A for any A ⊂ Ω. The advantage of this notation will appear later. Equivalent representations of a BBA m exist. In particular the implicability and commonality functions are defined, respectively, as: