Idempotent merging of Belief Functions: Extending the Minimum Rule of Possibility Theory Sebastien Destercke Didier Dubois Centre de cooperation internationale en Institut de Recherche en Informatique de Toulouse (IRIT) recherche agronomique pour le developpement (CIRAD) 118 Route de Narbonne, 31400 Toulouse UMR IATE, Campus Supagro, Montpellier, France Email: [email protected] Email: [email protected] Abstract—When merging belief functions, Dempster rule of combi- idempotence particularise [3] nation is justified only when information sources can be considered as P1; P2 P1 \P2 independent and reliable. When dependencies are ill-known, it is usual axioms [2], [5] Idempotent rule in to require the combination rule to be idempotent, as it ensures a cautious m1; m2 behaviour in the face of dependent sources. There are different strategies belief function frame idempotence to find such rules for belief functions. The strategy considered here π1; π2 min(π1; π2) generalise consists in relying on idempotent rules used in a more specific frameworks and to study its extension to belief functions. We study two possible extensions of the minimum rule of possibility theory to belief functions. Figure 1. Search of idempotent merging rules We first investigate under which conditions it can be extended to general contour functions.We then further investigate the combination rule that of belief functions. The idea is to request that the contour function maximises the expected cardinality of the resulting random set. after merging be the minimum of the contour functions of the input Keywords: least commitment, ill-known dependencies, contour belief functions. We first formulate into a strong requirement, and function, information fusion. then propose a weaker one as the former condition turns out to be too strong.Section IV studies the maximisation of expected cardinality as I. INTRODUCTION a practical tool for selecting a minimally committed merged belief When merging belief functions, the most usual rule to do so is structure. The notion of commensurate belief functions is used to gain Dempster’s rule of combination [4], normalized or not, which is insight as to the structure of focal element combinations allowing to justified only when the sources can be assumed to be independent. reach a maximal expected cardinality. This paper synthesises and There are other merging rules that assume a specific dependence completes previous results concerning our approach [6], [7] structure between sources [8], [17]. However, the (in)dependence II. PRELIMINARIES structure between sources is seldom well-kown. An alternative is then to apply the “least commitment principle”, which informally states We briefly recall basic tools needed in the paper. We denote by V that one should never presuppose more beliefs than justified. This the finite space on which the variable takes its values. principle is basic in the frameworks of possibility theory (minimal A. Belief functions, possibility distributions and contour functions specificity), imprecise probability (natural extension) [19], and the We assume our belief state is modelled by a belief function, or, Transferable Belief Model (TBM) [18]. It is natural to use it for the equivalently, by a basic belief assignment (bba). A bba is a function m cautious merging of belief functions. V P from the power set 2 of V to [0; 1] such that A⊆V m(A) = 1. We There are different approaches to cautiously merge belief functions, V denote by MV the set of bba’s on 2 . A set A such that m(A) > 0 but they all agree on the fact that a cautious conjunctive merging is called a focal set, and the value m(A) is the mass of A. This rule should satisfy the property of idempotence, as this property value represents the probability that the statement V 2 A is a correct ensures that the same information supplied by two sources will model of the available knowledge about variable V . We denote by remain unchanged after merging. There are three main strategies to F the set of focal sets corresponding to bba m. Given a bba m, construct idempotent rules that make sense in the belief function belief, plausibility and commonality functions of an event E ⊆ V setting. The first one looks for idempotent rules that satisfy certain are, respectively X X X desired properties and appear sensible in the framework of belief bel(E) = m(A); pl(E) = m(A); q(E) = m(A) functions [2], [5]. The second relies on the natural idempotent rule ;6=A⊆E A\E6=; E⊆A consisting of intersecting sets of probabilities and tries to express it in the particular case of belief functions [3]. The third approach, A belief function measures to what extent an event is directly sup- explored in this paper, starts from the natural idempotent rule in a ported by the information, while a plausibility function measures the less general framework, possibility theory, and tries to extend it. If we maximal amount of evidence supporting this event. A commonality denote m1; m2 two belief functions, P1; P2 two sets of probabilities, function measures the quantity of mass that may be re-allocated to a and π1; π2 two possibility distributions, the three approaches are particular set from its supersets. The commonality function increases summarized in Figure 1 below. We explore two different ways that when larger focal sets receive greater mass assignments, hence the extend the minimum rule (in the sense that the minimum rule is greater the commonality degrees, the less informative is the belief recovered when particularised to possibility distributions). function. Note that the four representations contain the same amount Section II recalls basics of belief functions and defines conjunctive of information [16]. merging in this framework. Section III then studies to what extent the In Shafer’s seminal work [16], no references are made to any minimum rule of possibility theory can be extended to the framework underlying probabilistic interpretation. However, a belief structure m can also be interpreted as a convex set Pm of probabilities [19] such Example 1. Consider the two belief structures m1; m2 on the domain that Bel(A) and P l(A) are probability bounds: Pm = fP j8A ⊂ V = fv1; v2; v3g X; Bel(A) ≤ P (A)g: Probability distributions are retrieved when F1 m1 F2 m2 only singletons receive positive masses. This interpretation is closer E11 = fv2g 0.5 E21 = fv2; v3g 0.5 to random sets and to Dempster’s view [4]. E12 = fv1; v2; v3g 0.5 E22 = fv1; v2g 0.5 A possibility distribution [10] is a mapping π : V! [0; 1] such These two random sets have the same contour function, while that π(v) = 1 for at least one element v 2 V. It represents incomplete m1 @pl m2 and m2 @q m1. And πm1 = πm2 . information about V .Two dual functions, the possibility and necessity As all these notions induce partial orders between belief structures, function, are defined as: Π(A) = supv2A π(v) and N(A) = 1 − Π(Ac). it can be desirable (e.g., to select a single least-specific belief struc- ture) to use additional criteria inducing complete ordering between The contour function πm of a belief structure m is defined as a belief structures. One of such criteria, already used to cautiously mapping πm : V! [0; 1] such that, for any v 2 V, merge belief functions [7], [14], is the expected cardinality of a πm (v) = pl(fvg) = q(fvg); belief structure m, denoted by jmj and whose value is jmj = P m(E)jEj; . It is equal to the cardinality of the contour with pl; q the plausibility and commonality functions of m. A belief E2F function π [13], that is structure m is called consonant when its focal sets are completely m X ordered with respect to inclusion (that is, for any A; B 2 F, we jmj = πm (v): (2) have either A ⊂ B or B ⊂ A). In this case, the information v2V contained in the consonant belief structure can be represented by the We can now define the notion of cardinality-based specificity: possibility distribution whose mapping corresponds to the contour -specificity m is said to be more -specific than m if and only function π(v) = P m(E): For non-consonant belief structures, C 1 C 2 v2E if we have the inequality jm j ≤ jm j and this relation is denoted the contour function can be seen as a (possibly subnormalized) 1 2 m v m and by m m if the above inequality is strict. The possibility distribution containing a trace of the original information, 1 C 2 1 @C 2 following proposition relates both π-inclusions and -specificity to easier to manipulate than the whole random set. C other inclusion notions B. Inclusion and information orderings between belief functions Proposition 1. Let m1,m2 be two random sets. Then, the following Inclusion relationships are natural tools to compare the informative implications holds: contents of set-valued uncertainty representations. There are many I m m ! m m extensions of classical set-inclusion in the framework of belief 1 @s 2 1 @π 2 II m m ! m m functions [9], leading to the definitions of x-inclusions, with x 2 1 @π 2 1 @C 2 III m m ! m m fpl; bel; q; s; πg. Let m and m be two bba defined on V. Inclusion 1 @s 2 1 @C 2 1 2 IV m m ! m v m between them can be defined as follow: 1 @pl 2 1 C 2 V m1 q m2 ! m1 v m2 fpl; q; πg-Inclusion m1 is said to be pl-included (resp. q- and @ C π-included) in m2 if and only if, for all A ⊆ V, pl1(A) ≤ pl2(A) C.