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Toward an axiomatic definition of conflict between belief functions Sébastien Destercke and Thomas Burger

Abstract—Recently, the problem of measuring the conflict between two bodies of evidence represented by belief functions has known a regain of interest. In most works related to this issue, Dempter’s rule plays a central role. In this paper, we propose to study the notion of conflict from a different perspective. We start by examining consistency and conflict on sets, and extract from this settings basic properties that measures of consistency and conflict should have. We then extend this basic scheme to belief functions in different ways. In particular, we do not make any a priori assumption about sources (in)dependency, and only consider such assumptions as possible additional information. Index Terms—Information fusion, evidence theory, conflict, dependence, transferable belief model.

I. I NTRODUCTION When getting some information from a source or combining the information provided by several sources, it often happens that the resulting information is partially conflicting, for different reasons (non-completely reliable sources, interpretation disagreement, . . . ). Being able to quantify this conflict is an important issue, as further information processing can depend on how much conflict is present. For example, fusion rules can adapt themselves to the amount of conflict [1] or some specific action can be triggered when conflict is too important (e.g., send back alarm to the user, discard unreliable sources). In this paper, we study the notion of conflict and its quantification when information is represented by belief functions. Belief functions generalise many other uncertainty representations, such as classical sets, probabilities or possibility distributions. They have been proposed as a solution to solve many problems involving uncertainty, such as state estimation [2], clustering [3] or multi-criteria decision analysis [4]. Theories based on belief functions provide general frameworks to represent uncertain information and reason with this information in presence of imprecision. They mix set-based and probabilistic-based tools to integrate imprecision in uncertainty representations. These frameworks rely on the notions of mass assignment and belief function to model uncertainty but, depending on the authors, the semantic interpretation of these objects varies. For instance, in most of the Theory of Evidence [5] and in the Transferable Belief Model (TBM) [6], the notion of uncertainty is related to an ill-known variable W that has a unique true value on a space Ω (finite here). We will adopt this latter interpretation of uncertainty as our main thread Sebastien Destercke is with CNRS, UMR Heudiasyc, Centre de recherche de Royallieu, Compiègne, France (email: [email protected]). Thomas Burger is with (1) CNRS, iRTSV (FR3425) (2) CEA / iRTSV / Biologie à Grande Échelle (3) INSERM (U1038) (4) Université de Grenoble, France (email: [email protected]).

for simplicity purposes. Yet, we will connect our results to another interpretation where mass assignments represent some imprecise knowledge about the probability distribution of a random variable [7], [8]. In the Theory of Evidence and in the TBM, the notion of conflict between pieces of information and its measurement play essential roles. In particular, its uses in merging rules is the matter of lively debates [1]. Recently, some researchers have questioned the validity of the usual conflict measure [9]– [11] (the mass given to the empty set after combination by Dempster’s rule). In many cases, they have answered these questions by complementing the usual measure with others. In this work, we aim at laying down some basic properties that a conflict measure between belief functions should follow. To do so, we take a rather different path to study the notion of conflict between belief functions. Two main ideas motivate this study and its results: First, the idea that the properties of conflict measures between belief functions should be based on properties that appear natural when measuring conflict between sets. Among other things, this means that when belief functions reduce to sets, the conflict measure should also reduce to the one used between sets, i.e. it should be a proper extension of conflict measures between sets. Hence, sets are here our starting point to define desirable properties of conflict measures. Second, the idea that a measure of conflict between sources should not depend a priori on a specific (in)dependence assumption between the sources. Indeed, it often happens that sources are not independent or that the dependency between sources is unknown (by lack of information). In this case, conflict measurements should not be based on unverified assumptions. This is coherent with the principle of least commitment that states that one should take into account the available information, and nothing else. This need to relax the independence assumption is also present in other works about conflict [9], [12] where properties related to idempotence are advocated, as idempotence is related to the notion of dependent sources when merging belief functions [13]–[16]. To reach our goal, and similarly to what is done in possibility theory, conflict between mass assignments is measured in two steps. First, we define in Section III the consistency degree of a single mass assignment from consistency degree on sets. Second, we investigate the notion of conflict between sets in Section IV and extend our results to conflict between mass assignments in Section V. This study leads us to three different propositions of conflict measures, depending on how set conflict is generalized to mass assignments. Finally, we connect our results to recent approaches and works related

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to conflict in Sections VI and VII. Necessary preliminaries about belief functions are recalled in Section II. This article extends a short paper [17] version with complementary results, examples, proofs and discussion on related works. II. P RELIMINARIES In this section, we introduce the basics about belief functions, their formalisms and interpretations. A. Basics of belief functions A mass assignment m over Ω is a mapping m : ℘(Ω)�→ [0, 1], with ℘(Ω) the power set of Ω and such that A∈℘(Ω) m(A) = 1. MΩ denotes the set of all mass assignments over Ω. A focal element A of m is a subset of Ω such that m(A) > 0. The set of focal elements of m is noted F. A mass assignment m is said to be normalised if m(∅) = 0. From m are defined several set-functions [5], the main ones being the belief, plausibility and commonality functions, respectively denoted Bel, P l and Q and such that, for any A ⊆ Ω: � � � Bel(A) = m(E), P l(A) = m(E), Q(A) = m(E) E⊆A,E�=∅

E∩A�=∅

E⊇A

(1) The contour function pl : Ω → [0, 1] of a mass assignment corresponds to its plausibility on singletons. Among all possible mass assignments, several specific cases are often defined. A mass assignment is called: • vacuous if m(Ω) = 1. It models total ignorance (P l(A) = 1 and Bel(A) = 0 for any A �= {∅, Ω}); • empty if m(∅) = 1. It models completely inconsistent information, from which it is impossible to draw any conclusion; • categorical if there is a set E ⊂ Ω such that m(E) = 1. Categorical mass assignments are equivalent to sets; • consonant if its focal elements are included in each others, that is for any A, B ∈ F either A ⊆ B or B ⊆ A. The plausibility (belief) measure of a consonant mass assignment is equivalent to a possibility (necessity) measure, meaning that all the information of a consonant mass assignment is contained in its contour function. The mass assignment and the set-functions it induces can be interpreted in different ways. In this paper, we focus on the interpretation corresponding to Shafer’s main interpretation of belief function [5], extensively taken over by Smets in his Transferable Belief Model [6]. In this view, the mass m(A) is the mass of belief exactly committed to the hypothesis {ω0 ∈ A}, where ω0 is the true but ill-known value of a variable W. Compared to Shafer seminal work, one particular asset of the TBM is that it allows the mass on the empty set to be nonnull, i.e. m(∅) �= 0. In the TBM original exposure, m(∅) is not related to conflict itself, but to the open-world assumption in which m(∅) quantifies the belief that the true value does not lie in Ω. To avoid ambiguities, we will speak of the singular (as opposed to statistical) interpretation of mass assignments when assuming that m models the uncertainty of a variable W having a unique but ill-known value ω0 .

In this paper, we do not discuss the interpretation of a resulting non-null m(∅) (e.g., it can be due to a disagreement between sources, to some unreliable sources, or to the fact that ω0 �∈ Ω). We focus on defining the notion of conflict, independently of the reason of its presence. Even if the singular interpretation is our main anchor, we will also try to connect our results with the second main interpretation of mass assignments, namely the one that considers belief and plausibility measures Bel and P l as bounds of an ill-known probability measure (noted P r) on Ω. Within such an interpretation, mass assignments are associated to the convex set of probabilities dominated by Bel: Pm := {P r(.) | ∀A ⊆ Ω, Bel(A) ≤ P r(A)}.

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m can then be seen, for example, as frequencies of imprecise observations. In this interpretation it makes poor sense to authorize m(∅) �= 0, since this corresponds to Pm = ∅. B. Combining belief functions A main origin of conflict is the merging of information coming from multiple sources not fully agreeing with each other. Indeed, combining multiple pieces of information modelled by belief functions is a problem that has been studied for a long time. Roughly speaking, there are three basic kinds of combinations [18]: conjunctive, disjunctive and compromise. Here, we consider that pieces of evidence are combined conjunctively, i.e. that all sources are reliable and that the true value lies within their conjunction1 . One of the most classical conjunctive combination between belief functions is unnormalised Dempster’s rule [6]. It assumes that sources of information are independent. Here, we work in a more general framework [15], [19], [20], where other dependency structures among the sources are considered. Given two mass assignments m1 and m2 on Ω, we consider that a conjunctive combination is made in two steps: 1) A joint mass assignment m : ℘(Ω) × ℘(Ω) → [0, 1] is built, such that, for any A or B ∈ ℘(Ω), � � m(A × B) = m1 (A), ; m(A × B) = m2 (B). B⊆Ω

A⊆Ω

(3) 2) � A mass m∩ : ℘(Ω) → [0, 1] such that m∩ (C) = A∩B=C m(A × B) is obtained. We speak of conjunctive combination whenever a combination follows these two steps. Hence unnormalised Dempster’s rule, which is retrieved by choosing m(A × B) = m1 (A)m2 (B) in step 1, is just one conjunctive combination among others. In the sequel, we denote by m⊕ the mass obtained by unnormalised Dempster’s rule. We denote by M12 the set of all possible mass assignments m∩ that can be obtained from the conjunctive combination of m1 and m2 , and call such masses conjunctive. The joint mass m encodes the dependence structure between the two sources m1 , m2 . The set of operators corresponding to the above procedure has the vacuous mass 1 Disjunctive and compromise operators are usually designed to produce non-conflicting results, and are of no interest here.

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assignment as neutral element, i.e. if m2 is vacuous, then M12 = {m1 } since Ω ∩ A = A for any A ⊆ Ω. Example 1: Consider two mass assignments on Ω = {ω1 , ω2 , ω3 } such that m1 ({ω1 , ω2 }) = 0.6,

m1 ({ω1 , ω3 }) = 0.4;

m2 ({ω2 , ω3 }) = 0.5,

III. C ONSISTENT MASS ASSIGNMENTS

m2 (Ω) = 0.5.

Table I summarises the constraints of the conjunctively m2 ({ω2 , ω3 }) m �2 (Ω) m∩

m1 ({ω1 , ω2 }) m∩ ({ω2 }) m∩ ({ω1 , ω2 }) = 0.6

m1 ({ω1 , ω3 }) m∩ ({ω3 }) m∩ ({ω1 , ω3 }) = 0.4

�

m∩

= 0.5 = 0.5

merged masses m∩ ∈ M12 . The convex set M12 therefore contains all the masses m∩ satisfying the linear constraints m∩ ({ω2 }) + m∩ ({ω3 }) = 0.5

m∩ ({ω1 , ω2 }) + m∩ ({ω1 , ω3 }) = 0.5; m∩ ({ω2 }) + m∩ ({ω1 , ω2 }) = 0.6 m∩ ({ω3 }) + m∩ ({ω1 , ω3 }) = 0.4 and the result of Dempster’s rule m⊕ ({ω1 , ω2 }) = m⊕ ({ω1 , ω2 }) = 0.3, m∩ ({ω3 }) = m∩ ({ω1 , ω3 }) = 0.2 is one solution among others. Note that all mass assignments in M12 are specialisations of both m1 and m2 . A mass m with F = {E1 , . . . , Eq } is a specialisation [21] of m� with F � = {E1� , . . . , Ep� } if and only if there exists a non-negative matrix G = [gij ] such that q �

gij = 1,

i=1

gij > 0 ⇒ Ei ⊆ Ej� , p � for i = 1, . . . , q, m� (Ej� )gij = m(Ei ). j=1

This relation is denoted by m �s m� and by m �s m� if there is at least a pair i, j such that gij > 0 and Ei ⊂ Ej . The term gij is the proportion of the focal element Ej� that "flows down" to focal element Ei . In other words, m1 is s-included in m2 if the mass of any focal element Ej� of m2 can be redistributed among subsets of Ej in m1 . Thus, the specialisation relation is a direct extension of the relation of inclusion between classical sets. Note also that m �s m� implies P l ≤ P l� . The two following properties [15] of mass assignments in M12 are instrumental in the rest of the paper. Proposition 1: ∀m∩ ∈ M12 and ∀A ⊆ Ω, we have P l∩ (A) ≤ min(P l1 (A), P l2 (A)).

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Proposition 2: ∀ω ∈ Ω, ∃m∩ ∈ M12 such that pl∩ (ω) = min(pl1 (ω), pl2 (ω)).

In the sequel, the notion of conflict follows from the notion of consistent mass assignments, defined in this section. We first consider simple sets (i.e., categorical mass assignments) to establish basic properties that consistency measures should follow, before extending the results to mass assignments. A. Consistency of sets

TABLE I J OINT MASSES m∩ ∈ M12 IN E XAMPLE 1.

for j = 1, . . . , p,

Finally, it is worthwhile to recall [22] that the set of probabilities Pm1 ∩ Pm2 corresponds to taking all masses in M12 such that m∩ (∅) = 0. That is, Pm1 ∩ Pm2 �= ∅ as soon as there exists a normalised mass in M12 .

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These properties show that masses in M12 are indeed conjunctive (Eq. (4)), and that for singletons, the minimum between the two marginal plausibilities is reached for at least one element of M12 (Eq. (5)).

When information is provided as a single set ω0 ∈ A, this information is consistent if and only if A �= ∅. Note that receiving the information A = ∅ is possible: the set of models induced by a set of inconsistent formulas expressed in propositional logic is empty (and a single information source may provide such a set). That is, if Σ is a set of formulas and if A(Σ) denotes the set of interpretations under which all formulas of Σ are true, A(Σ) �= ∅ if and only if Σ is a consistent knowledge base. In this case, either a set is consistent or it is not, and a degree of consistency φ can only take two values. It should obey the following properties: Property 1 (Bounded): A measure of consistency φ should be bounded, i.e. possess minimal and maximal values. Property 2 (Extreme consistent values): A measure of consistency φ should be maximal iff information is totally consistent, and minimal iff information is totally inconsistent. For simplicity, we assume from now on that a consistency degree has values included in the unit interval [0, 1]. Since only two situations can occur in the case of single sets, we can define the consistency degree as the function φ : ℘(Ω) → {0, 1} such that � 1 if A �= ∅ φ(A) = (6) 0 if A = ∅ This function satisfies Properties 1 and 2. We now extend it to mass assignments. B. Consistency of mass assignments To define a consistency degree on mass assignments, we must first define totally consistent and totally inconsistent information in terms of such mass assignments. As the mass assignment modelling the empty set is the empty mass assignment (i.e. m(∅) = 1), it is natural to associate it with a totally inconsistent information state. Totally consistent information (i.e. sets), on the other hand, can be (and has been) extended in two main ways. A first definition [15], [23] is the following: Definition 1 (Logical consistency):�A mass assignment m is logically consistent if and only if E∈F E �= ∅. That is, a mass assignment whose focal elements have a non-empty intersection. Such mass assignments are nor� malised, since if ∅ was a focal element, E∈F E = ∅. The next lemma characterizes these mass assignments in terms of contour functions.

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� Lemma 1: E∈F E �= ∅ ⇔ ∃ω�∈ Ω s.t. pl(ω) = 1 � Proof: ⇒: direct, since if ∃w ∈ E∈F E, then pl(ω) = E∩ω�=∅ m(E) = 1 ⇐: assume this is not true, pl(ω) = 1 and there are two focal elements A, B such that A ∩ B = ∅ (if there is only one focal element A, then m(A) = 1 and pl(ω) = 1 for any ω ∈ A). We reach a contradiction, since either ω �∈ A or ω �∈ B, and m(A) > 0, m(B) > 0 by definition. This means that a mass assignment is logically consistent if and only if its contour function is normalized. This form of consistency agrees with the singular interpretation, where one searches to know the true state of the world ω0 . A source is therefore totally consistent if it considers that at least one state of the world is totally plausible. Remark 1: Among logically consistent mass assignments, consonant ones play a particular role, as they display a stronger form of consistency. Indeed, the intersection of any pair of focal elements of a consonant mass assignment is still a focal element of this mass assignment (since if A ⊂ B, A∩B = A), which is not the case for logically consistent mass assignments in general. Also, consonant mass functions can be associated to hierarchical consistent knowledge bases [24]. The next definition provides a weaker form of consistency: Definition 2 (Probabilistic consistency): A mass assignment m is probabilistically consistent if and only if m(∅) = 0. The name probabilistic consistency comes from the fact that requiring m(∅) = 0 is equivalent to requiring that Pm �= ∅. Def. 2 is also in accordance with logic-based interpretation of belief functions [20] (and of convex sets of probabilities [25]). Defs. 1 and 2 provide two characterisations of totally consistent belief functions, each suggesting different measures of consistency for belief functions. The following measures of consistency φpl , φm from MΩ to [0, 1], such that: φpl (m) = max pl(ω),

(7)

φm (m) = 1 − m(∅)

(8)

ω∈Ω

do satisfy Property 2 for Defs. 1 and 2 respectively. Indeed, by Lemma 1, we have that Eq. (7) is maximal if and only if m is logically consistent, while by definition Eq. (8) is maximal as soon as m(∅) = 0. They also consider m(∅) = 1 as totally inconsistent belief functions, as in such a case φpl (m) = φm (m) = 0. Finally, when m is a categorical mass assignment, Eqs. (7) and (8) reduce to Eq. (6). At first sight, Def. 2 and Eq. (8) seem more adapted the imprecise probabilistic interpretation of belief functions than to the singular interpretation. However, we will see in next sections that the consistency degree given by Eq. (8) can be useful within the singular interpretation. Also, Eqs. (8) and (7) are linked by the following lemma. Lemma 2: Let m be a mass assignment, then � the inequality φpl ≤ φm holds, and φpl = φm if and only if E∈F \∅ E �= ∅. Proof: Inequality follows from the fact that, for any ω ∈ Ω, we have � � � pl(ω) = m(E) ≤ m(E) = m(E) − m(∅) ω∈E

E⊆Ω,E�=∅

E⊆Ω

=

1 − m(∅) = φm

Equality for a given ω is obtained if�and only if {E|ω ∈ E} = {E|E ⊆ Ω, E �= ∅}, hence if ω ∈ E∈F \∅ E. Remark 2: For consonant mass assignments, equality between φpl and φm always holds and corresponds to the consistency degree usually used in possibility theory (see, e.g. [26]). It is also coherent with inconsistency degrees in possibilistic logic [27]. IV. C ONFLICT BETWEEN SETS We can now proceed to define conflict measurements among sources providing mass assignments as their information. As in previous sections, we start by discussing conflict between simple sets to establish basic properties that conflict measures should follow, before studying in Section V possible extensions to mass assignments. As in possibility theory [26], we measure conflict between information sources as the inconsistency resulting from the conjunctive merging of information, where the inconsistency degree is the inverse of the consistency degree (i.e., 1 − φ(·)). This idea has a major advantage: as long as the combination operator is associative and commutative, the definition can be extended straightforwardly to any number N of sources. Let m1 , . . . , mN be N mass assignments, and m∩ a conjunctive combination of these masses. Then, we formally define a conflict measure κ(.) as an application defined on MΩ , and κ(m∩ ) as the conflict between m1 , . . . , mN . As the binary case will allow for a clearer exposure of our idea, we focus on it and consider the measure of conflict κ(., .) as an application defined on MΩ × MΩ including the combination step. In the case of sets, only two situations may occur when two sources respectively say about a variable W that ω0 ∈ A and ω0 ∈ B: either they are conflicting (A ∩ B = ∅) or they are not (A ∩ B �= ∅). We have again two extreme situations: total conflict and non-conflict. As for the consistency measure, a (bounded) measure of conflict κ : ℘(Ω) × ℘(Ω) → [0, 1] should take its maximal and minimal values in such cases, and we can reformulate Property 2 as follows: Property 3 (Extreme conflict values): A measure of conflict should be maximal iff sources are totally conflicting, and be minimal iff sources are non-conflicting. In other words, conflict κ for sets should be such that � 1 if A ∩ B = ∅ κ(A, B) = 1 − φ(A ∩ B) = (9) 0 if A ∩ B �= ∅. Other desirable properties that a measure of conflict should have may be formulated by observing sets. A first property should be symmetry, as we consider the two sources to be of equal importance. Property 4 (Symmetry): A measure of conflict should be symmetric. This corresponds to requiring that κ(A, B) = κ(B, A). Most other properties (including the next one) concern the behaviour of the measure with respect to changes in the information. Property 5 (Imprecision monotonicity): A measure of conflict should be non-increasing if the imprecision of a source of information increases.

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If A∩B �= ∅, then replacing A by A� ⊇ A (a more imprecise information) implies A� ∩ B �= ∅, hence κ should not increase. In contrast, we may have A ∩ B = ∅ but A� ∩ B �= ∅, in which case κ should decrease. This translates by the constraint κ(A� , B) ≤ κ(A, B). Property 6 (Ignorance is bliss): A measure of conflict should be insensitive to combination with ignorance. More formally, if B = Ω, then A ∩ B �= ∅ unless A = ∅. Indeed, a state of ignorance should not conflict with any other state of information (modelled by sets in this section and by mass assignments in Section V), but this latter state of information may be partially inconsistent. This translates by the constraint κ(A, Ω) = 1 − φ(A). Before introducing the next property, recall that a refinement of a space Ω into a space Θ consists in "splitting" some elements of Ω in multiple elements of Θ. A refinement can be modeled by a function ρ : ℘(Ω) → ℘(Θ) such that: 1) The set {ρ({ω})|ω�∈ Ω} is a partition of Θ; 2) ∀A ⊆ Ω, ρ(A) = ω∈A ρ({ω}). Property 7 (Insensitivity to refinement): A measure of conflict should be insensitive to refinement. This property comes from the fact that if ρ is a refinement of Ω into Θ, then ρ(A)∩ρ(B) = ∅ if and only if A∩B = ∅. A similar property called equivariance with respect to vacuous extension [13] or resistance to refinement [1] is often required for combination rules. It is possible to extend Property 7 to the coarsening of a space Θ by considering a partition Ω of Θ, but the problem is that the image of a set A ⊂ Θ is not uniquely defined [5, p. 117] in this case. However, we can require the following: let A be the outer approximation of A in Ω, i.e., A = {ω ∈ Ω|ρ({ω}) ∩ A �= ∅}, then a measure of conflict should be non-increasing after coarsening. This translates into constraints that if A, B ⊆ Ω, then κ(A, B) = κ(ρ(A), ρ(B)) and that if A, B ⊆ Θ, then κ(A, B) ≤ κ(A, B). For example, let Ω = {red (r.), blue (b.), green (g.)} and Θ = {light red (l.r.), dark red (d.r.), b., g.} with ρ({r.}) = {l.r., d.r.}, ρ({g.}) = {g.} and ρ({b.}) = {b.}. If A = {r.} and B = {r., g.} are non-conflicting, then so are ρ(A) = {l.r., d.r.} and ρ(B) = {l.r., d.r., g.}. In contrast, A = {l.r.} and B = {d.r.} are conflicting, but A = B = {r.} are not. V. C ONFLICT BETWEEN MASS ASSIGNMENTS We now consider that sources provide mass assignments m1 , m2 . In this case the conjunctive combination is no longer unique (after Eq. (3)), unless a specific (in)dependence assumption can be made. In our opinion, conflict should not be measured according to a (in)dependence assumption unless the assumption is supported by available evidence. Especially, unnormalised Dempster’s rule should not be used unless independence between sources holds. This results in the following property Property 8 (Independence to dependence): A conflict measure should not depend on a dependence assumption not supported by evidence. Remark 3: In possibility theory, the min t-norm [28] is often used to compute the merged possibility distribution from which conflict is measured. However, this distribution is

less specific (in the sense of specialisation) than distributions obtained with any other t-norm. The min t-norm therefore corresponds to the principle of minimal commitment, which is not the case of the unnormalised Dempster’s rule. A. Characterising total conflict and non-conflict Characterising totally conflicting sources is easy: according to our conjunctive fusion scheme, two sources are totally conflicting if none of their focal elements intersect (i.e., nonempty sets cannot receive a positive mass). Let Di = ∪A∈Fi A denote the disjunction of every focal element of mi , then totally conflicting masses m1 and m2 are defined as follows Definition 3 (Total conflict): m1 and m2 are totally conflicting when D1 ∩ D2 = ∅. When m1 , m2 are categorical, Def. 3 corresponds to an empty intersection of sets. When sources provide mass assignments m1 , m2 , we see three main ways to extend the condition of non-conflicting sets (in the sense that we retrieve the condition A ∩ B �= ∅ when m1 , m2 are categorical), that we define from the most to the least restricting. Definition 4 (Strong non-conflict): Two sources are strongly non-conflicting iff � A �= ∅ A∈{Fm1 ∪Fm2 }

Definition 5 (Non-conflict): two sources are non-conflicting iff ∀A, B such that A ∈ Fm1 , B ∈ Fm2 , we have A ∩ B �= ∅. Definition 6 (Weak non-conflict): two sources are weakly non-conflicting iff Pm1 ∩ Pm2 �= ∅. Def. 4 requires that all focal elements of m1 and m2 have a non-empty intersection. It is stronger than asking each focal element of m1 to have a non-empty intersection with each focal element of m2 (Def. 5). Since Def. 5 implies that m∩ (∅) = 0 for every m∩ ∈ M12 (see [15] ), it is stronger than Def. 6 that is satisfied as soon as one m∩ ∈ M12 is such that m∩ (∅) = 0. All of them reduce to the condition of non-empty intersection for sets when m1 , m2 are categorical. Similarly to consistency, we can associate these conditions to various features of mass assignments. The first proposition shows that Def. 4 (Strong non-conflict) is related to the contour functions, and therefore to the consistency measure given by Eq. (7). Proposition 3: � A �= ∅ ⇔ ∃ω ∈ Ω s.t. ∀m∩ ∈ M12 , plm∩ (ω) = 1 A∈{Fm1 ∪Fm2 }

Proof: ⇒: by assumption, ∃ω s.t. for any A ∈ Fm1 , B ∈ Fm2 , ω ∈�(A ∩ B). Hence,�for any m∩ ∈ M12 , we have pl∩ (ω) = ω∈C m∩ (C) = ω∈A∩B m(A × B) = 1 �⇐: for all m∩ ∈ M12 , we have pl∩ (ω) = A∈F1 ,B∈F2 m∩ (A ∩ B). As this is true for all m∩ , we have ω ∈ A ∩ B for all A ∈ F1 , B ∈ F2 (since if A, B are focal elements, there is at least one m∩ ∈ M12 for which m∩ (A ∩ B) > 0). As ω ∈ A ∩ B ⇒ ω ∈ A, ω ∈ B, we have ω ∈ A for all A ∈ F1 and ω ∈�B for all B ∈ F1 . As ω is in every focal element, we have A∈{Fm ∪Fm } A �= ∅ 1

2

6

This suggests to use the contour function to evaluate the conflict under Strong non-conflict (Def. 4). Proposition 3 says that two sources are strongly non-conflicting if and only if there exists at least a common state of the world ω that they both consider "normal" or totally plausible. This is in agreement with the singular interpretation of mass assignments, where there is only one (ill-known) true state of the world. Concurring suggestions can be found in the literature [10]. Defs. 5 and 6 of non-conflict, on the other hand, are closely related to the consistency measure given by Eq. (8). Proposition 4: A∩B �= ∅ ∀A ∈ Fm1 , B ∈ Fm2 ⇔ m∩ (∅) = 0 ∀m∩ ∈ M12 � Proof: ⇐: just notice that m∩ (∅) = A∩B=∅ m(A × B) and that there is no such pair by assumption. ⇒: simply observe that if m∩ (∅) > 0, it means that there exists a pair A ∈ Fm1 , B ∈ Fm2 such that A ∩ B = ∅, which is impossible by assumption. This suggests to use m∩ (∅) as a measure of conflict under non-conflict (Def. 5). Although it seems less adapted to the singular interpretation, we will discuss in the sequel its possible use within this latter interpretation. It has also been used in works related to logical interpretation of mass assignments [20]. The next example shows that non-conflict (Def. 5) and strong non-conflict (Def. 4) are distinct and can lead to different conclusions. We will come back to this difference when considering how to measure conflict. Example 2: Consider the mass assignments m1 , m2 of Example 1. They satisfy Def. 5 but not Def. 4. The highest plausibility value of a singleton in M12 is pl∩({ω2 }) = 0.6 (it is reached, for example, by taking m∩ ({ω3 }) = m∩ ({ω2 }) = 0.25, m∩ ({ω1 , ω2 }) = 0.35 and m∩ ({ω1 , ω3 }) = 0.15). pl∩ ({ω3 }) and pl∩ ({ω1 }) are bounded by pl1 ({ω3 }) = 0.4 and pl2 ({ω1 }) = 0.5, respectively. This maximal joint plausibility lower than one shows that although both sources consider one element to be a totally plausible state of the world, they disagree on which one it is (ω1 for source 1, ω2 for source 2). Finally, we examine weak non-conflict (Def. 6). In this case, results from Chateauneuf [22] gives the following proposition Proposition 5: Pm1 ∩ Pm2 �= ∅ if and only if there exists a mass assignment m∩ ∈ M12 such that m(∅) = 0. This corresponds to the interpretation of belief functions as ill-known probabilities. In fact, the set Pm1 ∩Pm2 is equivalent to the set of mass assignments in M12 that gives a null mass to the empty set [22]. Here again, the proposition suggests to use m(∅) as a measure of conflict under weak non-conflict (Def. 6), but in a weaker sense than for non conflict. According to Def. 6 m1 , m2 are non-conflicting as soon as one mass m∩ is such that m∩ (∅) = 0, while Def. 5 requires every mass m∩ in M12 to give null mass to the empty set. This leads us to the following corollary: Corollary 1: m1 , m2 strongly non-conflicting (Def. 4) =⇒ m1 , m2 non-conflicting (Def. 5) =⇒ m1 , m2 weakly nonconflicting (Def. 6). These implications are strict. Proof: The counter-example given in Example 1 shows that the first implication is strict, while strictness of the second implication follows from Proposition 5.

B. Measuring conflict between mass assignments Using results and properties from the previous section, we propose different measures of conflict for each condition of non-conflict. We define conflict as the inconsistency resulting from conjunctive combination. If the dependence structure cannot be uniquely identified, then the result of conjunctive merging will be a set of joint masses from M12 , to which will correspond a set of conflict values, typically an interval. This interval will reflect our lack of knowledge in the dependence structure. We first reformulate properties of conflict measurement κ : MΩ × MΩ → [0, 1] in terms of mass assignments: • Prop. 3 (Extreme conflict values): κ(m1 , m2 ) = 0 if and only if m1 and m2 are non-conflicting (according to the considered definition); • Prop. 4 (Symmetry): κ(m1 , m2 ) = κ(m2 , m1 ); � • Prop. 5 (Imprecision monotonicity): if m1 �s m1 , then � κ(m1 , m2 ) ≤ κ(m1 , m2 ); • Prop. 6 (Ignorance is bliss): if m2 is vacuous, then κ(m1 , m2 ) = 1 − φ(m1 ); • Prop. 7 (Insensitivity to refinement): If ρ is a refinement function from Ω into Θ, define the refined mass assignment m1 as mρ(1) , such that for any focal element E ∈ F1 , we have m1 (E) = mρ(1) (ρ(E)). Then, Proposition 7 reads κ(m1 , m2 ) = κ(mρ(1) , m2 ). 1) Measures for strong non-conflict: given Proposition 3, we use the measure of consistency φpl (Eq. (7)) to measure conflict corresponding to Definition 4 (Strong non-conflict) and Definition 3. This gives us three possible cases: 1. the case where dependence is unknown, and where one accepts to have a set of possible conflict values. In this case, if we note I([0, 1]) the set of intervals of [0, 1], the measure of conflict κ1pl : MΩ × MΩ → I([0, 1]) is κ1pl (m1 , m2 ) = [ min 1 − φpl (m∩ ), max 1 − φpl (m∩ )] m∩ ∈M12

m∩ ∈M12

(10)

= [ min 1 − max pl∩ (ω), max 1 − max pl∩ (ω)] m∩ ∈M12

ω∈Ω

m∩ ∈M12

ω∈Ω

2. the case where dependence is unknown, but the principle of least commitment is followed to get a unique conflict value. In this case, we select the minimal conflicting situation (as done in possibility theory and in [20]), i.e. to take as conflict value κ2pl : MΩ × MΩ → [0, 1] such that κ2pl (m1 , m2 ) = min 1 − φpl (m∩ ) = min 1 − max pl∩ (ω) m∩ ∈M12

m∩ ∈M12

ω∈Ω

(11) 3. the case where dependence is known (i.e., a joint mass m is specified) and where the result of conjunction is m∩ . we propose to use κ3pl (m1 , m2 ) = 1 − φpl (m∩ ) = 1 − max pl∩ (ω) ω∈Ω

(12)

All these measures satisfy Properties 3 to 7, and do consider the case of unknown dependence (partially known dependence can be handled by substituting M12 with the subset of possible merged mass assignments in Eqs. (10) and (11)). Computations of lower and upper bounds of κ1pl are easy to perform. They respectively come down to maximize and

7

� minimize the values pl∩ (ω) = ω∈E m∩ (E) (a linear equation) under the linear constraints given by Eq. (3), hence to solve linear programs. The lower bound is even easier to get, as we have maxm∩ ∈M12 pl∩ (ω) = min(pl1 (ω), pl2 (ω)) (after Eq. (4) and (5)) and minm∩ ∈M12 1 − maxω∈Ω pl∩ (ω) = 1 − maxω∈Ω maxm∩ ∈M12 pl∩ (ω). Example 3: Consider the two mass assignments of Example 1. We get the following results: κ1pl (m1 , m2 )

= [0.4, 0.4] =

κ2pl

=

κ3pl

In this case, the conflict measure has a unique value indicating average conflict. This is due to the fact that in Table I each singleton appears only in one column or one line. Example 4: Consider now two identical masses m1 , m2 on Ω = {ω1 , ω2 }: m1 ({ω1 }) = m2 ({ω1 }) = 0.5; m1 ({ω2 }) = m2 ({ω1 }) = 0.5. In this case, we obtain the following result: κ1pl (m1 , m2 ) = [0.5, 1], κ2pl (m1 , m2 ) = 0.5. κ1pl indicates some conflict that can be total (obtained by selecting m(ω1 × ω2 ) = 0.5 and m(ω2 × ω1 ) = 0.5 in Eq. (3)) or reduced to 0.5 if the merging rule is idempotent (m(ω1 × ω1 ) = 0.5 and m(ω2 × ω2 ) = 0.5 in Eq. (3)). In this example the two masses are identical and it is reasonable to assume they could agree together (at least when dependence is unknown). Hence the minimal conflict of 0.5 does not come from a disagreement between the sources, but rather from the self-inconsistency of the information provided by each source. This matter is discussed in Section VI. 2) Measures for non-conflict: given the link of Proposition 4 with Def. 2, we propose to use consistency measure φm (Eq.(8)) to measure conflict according to Defs. 5 (Nonconflict) and 3 (Total conflict). Again, we define 3 measures: 1. in the case where dependence is unknown and conflict imprecisely valued, we have κ1m : MΩ × MΩ → I([0, 1]) such that

Cattaneo [13], [20] measure is κ2m ; hence our measures for non-conflict are coherent with these two approaches. Example 5: For the mass assignments of Example 1, we get κ1m (m1 , m2 ) = [0, 0] since m1 , m2 are non-conflicting under Def. 5 (all focal elements of m∩ in Table I are non-empty). Example 6: Consider the masses of Example 4. In this case, we get the following results: κ1m (m1 , m2 ) = [0, 1]; κ2m (m1 , m2 ) = 0 The conclusion is that the sources could be totally conflicting (case where m(ω1 × ω2 ) = 0.5 and m(ω2 × ω1 ) = 0.5) or not at all if the merging rule is idempotent (case where m(ω1 × ω1 ) = 0.5 and m(ω2 × ω2 ) = 0.5), that is if they are completely dependent and agree together. In contrast with Example 4, the lower bound of the conflict is nil, and we do identify that m1 , m2 are potentially agreeing with each other. This suggests that measures κ1m and κ2m are interesting, as they appear more adapted than κ1pl and κ2pl to measure the conflict resulting from the combination of sources. This point is further discussed in Section VII. Also note that the lower bound of κ1m is always zero when m1 and m2 are identical, the idempotent combination being included in M12 . 3) Measures for weak non-conflict: Weak non-conflict (Def. 6) already implicitly assumes some kind of ill-known dependencies, as intersection of sets of probabilities comes down to consider the sets of all normalised m∩ ∈ M12 . As two masses are non-conflicting as soon as there exists a normalised mass in M12 , using κ2m (m1 , m2 ) to measure conflict seems reasonable, as κ2m (m1 , m2 ) = 0 if and only if M12 contains a normalised mass. VI. I NTERNAL AND EXTERNAL CONFLICT

Example 4 indicates that in the case of a measure of conflict based on the contour function, the conflict may come from both the disagreement between the sources and the inconsistency of each source. If two sources give identical mass assignments, then in case of total dependence and agreement, one could expect a null conflict (as in Example 6). In this κ1m (m1 , m2 ) = [ min 1 − φm (m∩ ), max 1 − φm (m∩ )] section, we show that measures based on the contour function m∩ ∈M12 m∩ ∈M12 (13) can be linked to the notion of internal and external conflict. = [ min

m∩ ∈M12

m∩ (∅), max m∩ (∅)] m∩ ∈M12

2. in the case where one adopts the principle of leastcommitment, we get κ2m : MΩ × MΩ → [0, 1] such that κ2m (m1 , m2 ) = min 1 − φm (m∩ ) = min m∩ (∅) m∩ ∈M12

m∩ ∈M12

(14)

3. in the case where dependence is known (i.e., a joint mass m is specified), we propose to use κ3m (m1 , m2 ) = 1 − φm (m∩ ) = m∩ (∅)

(15)

All these measures do satisfy Properties from 3 to 7, and take account of unknown dependencies. The common measure of conflict m⊕ (∅) [5] is captured by κ3m (m1 , m2 ) when independence between sources can be assumed, while

A. Previous work In his works, Daniel [10], [29] introduces the idea of separating internal from external conflict to identify conflict origins. He proposes to separate the total conflict T C(m1 , m2 ) resulting from a conjunctive combination of two masses m1 and m2 into three parts: the internal conflict of m1 and m2 , respectively denoted IC(m1 ) and IC(m2 ), and an external conflict among m1 and m2 , EC(m1 , m2 ). The value IC(mi ) should quantify the (self-)inconsistency of the ith source, while EC(m1 , m2 ) is only based on the interaction between m1 and m2 and should not integrate any self-inconsistency. Based on this idea, some relations that should satisfy the three conflict parts are proposed (we refer to Daniel [10] for details). They can be summarized as the

8

conditions

We propose to use the following strategy:

T C(m1 , m2 ) ≥ max (EC(m1 , m2 ), IC(m1 ), IC(m2 )) ; (16) T C(m1 , m2 ) ≤ EC(m1 , m2 ) + IC(m1 ) + IC(m2 ). (17) B. A needed distinction? A first question is whether such a distinction is useful. In our opinion, the answer is yes, as indicates the next example. Example 7: Consider the mass assignments of Example 1. We have obtained the values κipl (m1 , m2 ) = 0.4 for i ∈ {1, 2, 3} but we also have that κim (m1 , m2 ) = 0 for i ∈ {1, 2, 3}, since m1 and m2 are non-conflicting. Now, changing m1 to m∗1 ({ω2 }) = 0.6, m∗1 ({ω3 }) = 0.4 does not change the conflict measures. However, the internal conflict of m∗1 (w.r.t. Def. 1) is clearly more important. This shows that making a distinction between internal and external conflict makes sense, as a global conflict measure may fail to capture a significant change in internal or external conflict. As argued before, measures derived from non-conflict (Def. 5) do not really allow to measure conflict within a mass assignment, unless one allows a single source to provide a mass assignment m with m(∅) > 0, and this seldom happens in practice. This means that they are unable to capture internal conflict. On the other hand, the notion of internal conflict is really linked to that of logical consistency of a source (Def. 1). This is why we propose to build internal and external conflict measures using measures based on the contour function (Def. 4). As we are using contour functions, the proposed separation essentially makes sense if mass assignments are considered within the singular interpretation. C. Measuring the different parts Assuming that the dependence structure is unknown, we propose to use T C(m1, m2) = κ2pl (m1 , m2 ) as a measure of total conflict (as computing single values seems more useful to make a distinction between internal and external conflict). As internal conflict measure of a mass assignment m, we simply propose to take IC(m) = 1 − φpl (m) = 1 − maxω∈Ω pl(ω): a consistent source should consider that at least one state of the world is fully plausible. Other measures of internal conflict have been proposed [30], [31], however they rely on unnormalised Dempster’s rule of combination, a feature we want to avoid. Separating external conflict from total conflict is a bit more complex. As Daniel [10] suggests, we propose to first transform original mass assignments to eliminate internal conflict and to compute external conflict from the transformed masses. The following condition expresses the idea that external conflict should be nil if and only if both sources agree on which state of the world is the most plausible. Property 9 (External agreement): If m1 , m2 are two normalised mass assignments, then EC(m1 , m2 ) = 0 if and only if arg max pl1 (ω) = arg max pl2 (ω) ω∈Ω

ω∈Ω

1) Transform m1 and m2 into consonant masses m�1 and m�2 that have null internal conflicts. 2) Compute EC(m1 , m2 ) = κ2pl (m�1 , m�2 ) = 1 − maxω∈Ω min(pl1� (ω), pl2� (ω)), with pl1� , pl2� being the plausibility functions of m�1 , m�2 . The next step is to find a suitable consonant transformation from mi to m�i . We show here that the normalisation of the contour functions satisfy both Property 9 and inequalities (16)(17). This transform is based on the following procedure: 1) Compute the contour functions pl1 , pl2 of m1 , m2 . 2) Define pl1� = pl1/α, pl2� = pl2/β with α = maxω∈Ω pl1 (ω) and β = maxω∈Ω pl2 (ω) 3) Compute m�1 , m�2 from pl1� , pl2� . This transform can be seen as the possibilistic counterpart of the plausibility transform [32], [33] that transforms a plausibility in a probability. Using it, we have: Proposition 6: Let m1 , m2 be two mass assignment, IC(m) = 1 − φpl (m), T C(m1 , m2 ) = κ2pl (m1 , m2 ) and EC(m1 , m2 ) = κ2pl (m�1 , m�2 ), then the two Inequalities (16) and (17) hold. Proof: Let us prove the first inequality, and start with T C(m1 , m2 ) ≥ max{IC(m2 ), IC(m1 )}. Since all m∩ ∈ M12 are specialisations of both m1 and m2 , we have pl∩ (ω) ≤ min{pl1 (ω), pl2 (ω)} for each ω ∈ Ω and all m∩ ∈ M12 . Therefore, T C(m1 , m2 ) ≥ max{IC(m1 ), IC(m2 )}. As α ≤ 1 and β ≤ 1, we also have pl1 ≤ pl1� and pl2 ≤ pl2� , hence for each ω, pl∩ (ω) ≤ min{pl1 (ω), pl2 (ω)} ≤ min{pl1� (ω), pl2� (ω)} from which we can conclude T C(m1 , m2 ) ≥ EC(m1 , m2 ). Thus, the first inequality is proved. Let us now prove the second inequality. First, as T C(m1 , m2 ) = 1 − maxω∈Ω min{pl1 (ω), pl2 (ω)}, we have 1−maxω∈Ω min{pl1 (ω), pl2 (ω)} ≤ 1−min{pl1 (ω � ), pl2 (ω � )} for any ω � ∈ Ω. Also, assuming that α = maxω∈Ω pl1 (ω) and β = maxω∈Ω pl2 (ω), we have IC(m1 ) = 1 − α and IC(m2 ) = 1 − β. Consider now the element ω ∈ Ω such that ω = arg maxω∈Ω min(pl1� (ω), pl2� (ω)) (that is, the ω used to give its value to EC(m1 , m2 )). We have EC(m1 , m2 ) = 1 − min(pl1 (ω)/α, pl2 (ω)/β ). Assume, without loss of generality, that pl1 (ω) ≤ pl2 (ω). Two cases can happen: Either pl1 (ω)/α ≤ pl2 (ω)/β or pl1 (ω)/α ≥ pl2 (ω)/β . We will show that, whatever the case, 1 − pl1 (ω) (i.e. the upper bound of T C(m1 , m2 )) is dominated by EC(m1 , m2 ) + IC(m1 ) + IC(m2 ). •

If

pl1 (ω)/α

≤ pl2 (ω)/β , then, we have

1 − pl1 (ω) ≤ 1 − α + 1 − β + 1 −

•

pl1 (ω) α

since α(1 − pl1 (ω)) ≤ α(1 − α + 1 − β) + 1 − pl1 (ω) is true, as α(1 − pl1 (ω)) ≤ 1 − pl1 (ω) and (1 − α + 1 − β) is positive. If pl1 (ω)/α ≥ pl2 (ω)/β , then, we have 1 − pl1 (ω) ≤ 1 − α + 1 − β + 1 −

pl2 (ω) . β

Indeed, we have 1 − pl1 (ω) ≤ 1 − α (pl2 (ω)/β ) and we have pl2 (ω) pl2 (ω) 1−α ≤1−α+1−β+1− . β β The last inequality holds: moving everything to the righthand side we get 0 ≤ −α + α (pl2 (ω)/β ) + 1 − β + 1 − (pl2 (ω)/β ) from which we have 0 ≤ (1 − β) + (1 − α)(1 −

pl2 (ω) ), β

and all terms between parenthesis are positive. We therefore have that 1−pl1 (ω) ≤ EC(m1 , m2 )+IC(m1 )+ IC(m2 ), which finishes the proof. Let us revisit Example 7 using the above results: Example 8: Consider the mass assignments m1 , m2 of Example 1, and m∗1 , the modified m1 from Example 7. The conflict is then decomposed in the following way: T C(m1 , m2 ) = 0.4, IC(m1 ) = 0, EC(m1 , m2 ) = 0.4; T C(m∗1 , m2 ) = 0.4, IC(m∗1 ) = 0.4, EC(m∗1 , m2 ) = 0, and IC(m2 ) = 0 in both cases. By separating the conflict components, we are able to better identify its origins. Other transformations of m1 and m2 into consonant masses are explored in Appendix A, where it is shown that they do not satisfy Eqs. (16) and (17) or Property 9. Note that the proposed EC does not include any information about the dependency, hence our proposal essentially makes sense in case of unknown dependence. Including dependencies in the external conflict is the matter of further research. VII. D ISCUSSION ON RELATED WORKS This section briefly discusses the links of the present work with other conflict measurement approaches. A. Empty set mass or singleton plausibilities: the case of independence In the current framework, measures of conflict adapted to the singular interpretation are κipl , i ∈ {1, 2, 3}, depending on the (in)dependence assumptions. However, in many applications the conflict is measured according to κ3m , and more precisely according to m⊕ (∅), irrespectively of the interpretation of m and of the available information about source dependence. Hence, one may wonder to which extent the choice will affect results. To provide some insight about the practical impact of this choice, we made the following experiment: given a frame Ω of fixed size, we have drawn 10,000 samples of a fixed number of n normalised masses (drawn uniformly and independently from the set of all possible masses [34]). We combined them according to unnormalised Dempster’s rule (as masses are drawn independently) and computed κ3pl and κ3m for each sample. In this case κ3pl = 1 − maxω∈Ω pl⊕ (ω) is very easy to compute, as pl⊕ (ω) = pl1 (ω)pl2 (ω) (it can even be evaluated without performing the combination).

Nb. of sources

9

2 3 4 5 6 7

2 0.852 0.904 0.927 0.945 0.957 0.966

Size of frame Ω 3 4 5 0.779 0.721 0.663 0.833 0.768 0.714 0.852 0.788 0.729 0.876 0.812 0.745 0.896 0.821 0.764 0.907 0.832 0.770

6 0.605 0.674 0.689 0.701 0.698 0.718

7 0.558 0.624 0.637 0.652 0.655 0.658

TABLE II C ORRELATION OF κ3pl AND κ3m

Table II shows correlations obtained from the simulation. They are always high and increases with the number of masses n or as the frame Ω size decreases. In the first case, this is due to the fact that both measures tends to 1 as n increases, and in the second case this is due to the fact that the number of possible masses exponentially increases with Ω size. This correlation may even be higher in case of source dependence: in a word recognition problem, we combined the consonant mass functions given by three evidential classifiers [35] trained on the same data sets (hence dependent) with unnormalized Dempster’s rule, and computed κ3m and κ3pl . Two data sets (each counting 3000 items) were used, and the computed correlations were 0.82 and 0.85. These results clearly indicate that, even if κ3pl should be chosen in a singular interpretation, the correlation between m⊕ (∅) (or κ3m in general) and κ3pl is high enough to consider κ3m as a good approximation for κ3pl . B. Distance-based conflicts Liu [9] argues that using the mass m⊕ (∅) resulting from Dempster’s unnormalised rule as the unique measure of conflict may be misleading when assessing whether two sources agree together. To solve this situation, she proposes to use a bivariate conflict measure where the second component is the supremum norm dif f BetP of the pignistic transform. Jousselme et al. [36] show that dif f BetP is a particular measure of distance between mass assignments, and other authors have proposed conflict measures based on distances between mass assignments (e.g., see Martin et al. [11]). We agree with Liu and others that m⊕ (∅) should not be used to measure conflict in all situations, and that summarising every aspect of conflict to a single measurement may be difficult. However, we disagree on the fact that a measure of distance d(m1 , m2 ) between mass assignments should be used either as a unique measure or as a complementary measure of conflict, for the following reasons: - the maximal distances will not necessarily be obtained for situations of total conflict, hence not satisfying Property 3 (Extreme conflict values); - distance measures will be sensitive to the frame Ω size, hence will not satisfy Property 7 (Insensitivity to refinement); - Property 5 (Imprecision monotonicity) will in general not be guaranteed: an example is given when one takes two identical masses m1 and m2 with distance d(m1 , m2 ) = 0. Then if m1 is replaced by m�1 such that m1 �s m�1 , we get d(m�1 , m2 ) > 0 as m�1 , m2 are different; - Property 6 (Ignorance is bliss) will usually not be satisfied, as the vacuous mass assignment has a non-null distance from any non-vacuous mass assignment;

10

- extending distances to more than 2 masses is not straightforward, and existing proposals [11] usually come back to pair-wise distances by some means. Martin [12] recently proposed to combine distance measurement with a measure of inclusion, showing that it satisfies a number of desirable properties. In particular, it satifies Property 6 and a very strong version of Property 3, yet it is still sensitive to the frame Ω definition. Distance-based measures are very useful as practical tools to measure differences between mass assignments: the above remarks just point out that they should not be confused with conflict measurements. C. Illustrative application to Liu’s rule selection framework As an illustration of how our measures can be used, we can apply our framework to Liu’s idea of deciding whether it is safe to use a conjunctive merging approach, given some conflict evaluation. As already argued, distances do not fit our framework, and unnormalised Dempster’s rule should be used only if independence can be assumed. Hence, we would use the measure κim rather than a couple (m⊕ (∅), d(m1 , m2 )) to evaluate the safeness of using a conjunctive merging. Since we do not assume any dependency structure a priori, proposed measures of conflict can only help in deciding whether it is safe to combine conjunctively or not (according to the scheme of Section II-B), and whether independence holds or not should be assessed by other means. Although combination of evidence mostly concerns the singular interpretation, we prefer κim rather than κipl , despite the fact that κipl fits better with the singular interpretation. The reason is that κipl includes both internal and external conflicts, and the choice of a combination strategy should only (or mainly) rely on external conflict. Another solution would be to use a decomposition of κipl similar to the one of Section VI that can cope with independence information, however developing such a solution is out of the scope of the current work. Also, results of Section VII-A suggest that κim can serve as an approximation of κpl . As Liu [9], we propose to set two conflict thresholds �1 < �2 such that if conflict is above �2 , one should not use a conjunctive approach, and if conflict is below �1 , then conjunctive approaches can be safely used. We use either κ1m (dependency unknown) or κ3m (dependency known) to measure the conflict, and propose the following guidelines: 1 3 • do not use conjunctive approach if �2 ≤ κm or �2 ≤ κm ; 1 • be cautious with conjunctive approach if ]�1 , �2 [∩κm �= ∅ or κ3m ∈]�1 , �2 [; 1 3 • use conjunctive approach if �1 ≥ κm or �1 ≥ κm ; 1 where �2 ≤ κm means that the lower bound of interval κ1m is at least as high as �2 , and �1 ≥ κ1m means that the upper bound of interval κ1m is at least as low as �1 . Example 9: Let �1 = 0.4 and �2 = 0.7, and consider the mass assignments m1 , m2 over Ω = {ω1 , ω2 , ω3 , ω4 } such that m1 ({ω1 , ω2 }) = 0.8, m2 ({ω1 , ω2 }) = 0.1,

m1 ({ω3 }) = 0.1, m2 ({ω3 }) = 0.1,

m1 ({ω4 }) = 0.1, m2 ({ω4 }) = 0.8.

In this case, we have κ1m (m1 , m2 ) = [0.7, 1], and a conjunctive merging rule should not be used. The lower bound is reached by setting the joint mass m({ω3 } × {ω3 }) = 0.1, m({ω4 } × {ω4 }) = 0.1 and m({ω1 , ω2 } × {ω1 , ω2 }) = 0.1 in Eq. (3), while the upper bound is reached by setting m({ω1 , ω2 } × {ω3 }) = 0.1, m({ω4 } × {ω1 , ω2 }) = 0.1 (for each bound, the rest of the mass can be assigned arbitrarily). Example 10: Keeping the same thresholds �1 , �2 and Ω as in the preivous example, let m1 , m2 be such that m1 ({ω1 , ω2 , ω4 }) = 0.8, m1 ({ω3 }) = 0.1, m1 ({ω4 }) = 0.1, m2 ({ω1 , ω2 }) = 0.1,

m2 ({ω3 }) = 0.1,

m2 ({ω4 }) = 0.8.

In this case, we have κ1m (m1 , m2 ) = [0, 0.3], and using a conjunctive merging rule is ok, as the sources largely agree on the fact that ω4 can be the true value. When concluding that a conjunctive approach is not safe, actions to take (selecting another rule, discounting or rejecting some sources, etc.) depend on the available information (whether sources reliabilities are known, etc.) and on the application (maximizing classification accuracy, preventing from some risks, etc.). Similarly, setting the values �1 , �2 largely depends on the context. VIII. C ONCLUSION Our goal was to study the notion of conflict in belief function frameworks from a new perspective and to suggest basic properties for conflict measures. We have started from observations on simple sets, the most basic framework to represent uncertainty, and have extracted some properties of conflict measures from these observations. From these properties, we first defined two notions of consistent mass assignments and proposed consistency measures from these definitions. We have then defined conflict as the inconsistency arising from a conjunctive combination. Following these properties, we defined different notions of non-conflicting mass assignments and proposed conflict measurements from these notions, making no a priori assumptions about the dependence between sources. This led us to define interval-valued conflict when the dependence structure is ill-known (eventually coming back to one measure using the least-commitment principle). In several cases and under specific assumptions, we retrieve some existing measures, indicating that our framework encompass these latter measures. Roughly speaking, we specified two families of conflict measure: one based on the contour function, κipl , the other based on the mass assigned to the empty set, κim . While the latter is the most commonly used, our results indicate that only the former really fits with the singular interpretation. However κim is not without utility, and in some cases can be used as a good approximation of κipl . It has also a practical interest as a measure of conflict between sources. We have started to illustrate how our framework could be applied to some problems such as the separation of internal and external conflict or the determination of whether adopting a conjunctive behaviour is safe when merging sources of information. We can mention several perspectives to this work: refining the proposed framework to specific needs and application;

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evaluating the practical interest and implications of the proposed measures; comparing their numerical behaviours with those of other measures; applying the current framework and properties to the interesting problem of isolating conflict contribution between sources [11], [37]. R EFERENCES [1] P. Smets, “Analyzing the combination of conflicting belief functions,” Information Fusion, vol. 8, pp. 387–412, 2007. [2] G. Nassreddine, F. Abdallah, and T. Denoeux, “State estimation using interval analysis and belief-function theory: Application to dynamic vehicle localization,” IEEE Trans. on Syst., Man, and Cybern. B, vol. 40, no. 5, pp. 1205–1218, 2010. [3] T. Denoeux and M.-H. Masson, “Evclus: evidential clustering of proximity data,” IEEE Trans. on Syst., Man, and Cybern. B, vol. 34, no. 1, pp. 95–109, 2004. [4] M. Beynon, B. Curry, and P. Morgan, “The dempster–shafer theory of evidence: an alternative approach to multicriteria decision modelling,” Omega, vol. 28, no. 1, pp. 37 – 50, 2000. [5] G. Shafer, A mathematical Theory of Evidence. New Jersey: Princeton University Press, 1976. [6] P. Smets and R. Kennes, “The transferable belief model,” Artif. Intell., vol. 66, pp. 191–234, 1994. [7] E. Miranda, I. Couso, and P. Gil, “Random sets as imprecise random variables,” J. of Math. Analysis and Applications, vol. 307, 2005, 32-47. [8] A. Dempster, “Upper and lower probabilities induced by a multivalued mapping,” Ann. of Math. Stat., vol. 38, pp. 325–339, 1967. [9] W. Liu, “Analyzing the degree of conflict among belief functions,” Artif. Intell., vol. 170, no. 11, pp. 909–924, 2006. [10] M. Daniel, “Conflicts within and between belief functions,” in IPMU, 2010, pp. 696–705. [11] A. Martin, A.-L. Jousselme, and C. Osswald, “Conflict measure for the discounting operation on belief functions,” in The 11th International Conference on Information Fusion, 2008, pp. 1003–1010. [12] A. Martin, “About conflict in the theory of belief functions,” in The 2nd International Conference on Belief Functions, 2012. [13] M. E. G. V. Cattaneo, “Belief functions combination without the assumption of independence of the information sources,” Int. J. Approx. Reasoning, vol. 52, no. 3, pp. 299–315, 2011. [14] D. Dubois and R. Ronald, “Fuzzy set connectives as combinations of belief structures,” Information Sciences, vol. 66, no. 3, pp. 245–276, 1992. [15] S. Destercke and D. Dubois, “Idempotent conjunctive combination of belief functions: Extending the minimum rule of possibility theory,” Information Sciences, vol. 181, no. 18, pp. 3925 – 3945, 2011. [16] T. Denoeux, “Conjunctive and disjunctive combination of belief functions induced by non-distinct bodies of evidence,” Artificial Intelligence, vol. 172, pp. 234–264, 2008. [17] S. Destercke and T. Burger, “Revisiting the notion of conflicting belief functions,” in The 2nd International Conference on Belief Functions, 2012. [18] I. Bloch, “Information combination operators for data fusion : A comparative review with classification,” IEEE Trans. on Syst., Man, and Cybern. A, vol. 26, no. 1, pp. 52–67, January 1996. [19] D. Dubois and R. Yager, “Fuzzy set connectives as combination of belief structures,” Information Sciences, vol. 66, pp. 245–275, 1992. [20] M. Cattaneo, “Combining belief functions issued from dependent sources.” in Proc. Third International Symposium on Imprecise Probabilities and Their Application (ISIPTA’03), Lugano, Switzerland, 2003, pp. 133–147. [21] D. Dubois and H. Prade, “A set-theoretic view on belief functions: logical operations and approximations by fuzzy sets,” Int. J. of General Systems, vol. 12, pp. 193–226, 1986. [22] A. Chateauneuf, “Combination of compatible belief functions and relation of specificity,” in Advances in the Dempster-Shafer theory of evidence. New York, NY, USA: John Wiley & Sons, Inc, 1994, pp. 97–114. [23] F. Cuzzolin, “On consistent approximations of belief functions in the mass space,” in ECSQARU, 2011, pp. 287–298. [24] D. Dubois and H. Prade, “Possibilistic logic: a retrospective and prospective view,” Fuzzy Sets and Systems, vol. 144, no. 1, pp. 3–23, 2004. [25] G. de Cooman, “Belief models: An order-theoretic investigation,” Ann. Math. Artif. Intell., vol. 45, no. 1-2, pp. 5–34, 2005.

[26] D. Dubois and H. Prade, “Possibility theory and data fusion in poorly informed environments,” Control Eng. Practice, vol. 2, pp. 811–823, 1994. [27] S. Benferhat and H. Prade, “Compiling possibilistic knowledge bases.” in ECAI’06, 2006, pp. 337–341. [28] E. Klement, R. Mesiar, and E. Pap, Triangular Norms. Dordrecht: Kluwer Academic Publisher, 2000. [29] M. Daniel, “Non-conflicting and conflicting parts of belief functions,” in Seventh International Symposium on Imprecise Probability: Theory and Applications, 2011. [30] J. Schubert, “The internal conflict of a belief function,” in The 2nd International Conference on Belief Functions, 2012. [31] C. Osswald and A. Martin, “Understanding the large family of dempstershafer theory’s fusion operators - a decision-based measure,” in The 9th International Conference on Information Fusions, 2006. [32] F. Cuzzolin, “Credal semantics of bayesian transformations in terms of probability intervals,” IEEE Trans. on Syst., Man, and Cybern. B, vol. 40, no. 2, pp. 421–432, 2010. [33] B. R. Cobb and P. P. Shenoy, “On the plausibility transformation method for translating belief function models to probability models,” Int. J. Approx. Reasoning, vol. 41, no. 3, pp. 314–330, 2006. [34] T. Burger and S. Destercke, “Random generation of mass functions: A short howto,” in The 2nd International Conference on Belief Functions, 2012. [35] T. Burger, Y. Kessentini, and T. Paquet, “Dempster-shafer based rejection strategy for handwritten word recognition,” in Proceedings of the International Conference on Document Analysis and Recognition, (ICDAR), Beijing, China, September 18-21, 2011, 2011, pp. 528–532. [36] A.-L. Jousselme and P. Maupin, “Distances in evidence theory: Comprehensive survey and generalizations,” Int. J. of Approx. Reasoning, vol. In Press, Accepted Manuscript, 2011. [37] J. Klein and O. Colot, “Singular sources mining using evidential conflict analysis,” Int. J. of Approx. Reasoning, vol. 52, pp. 1433–1451, 2011. [38] P. Smets, “Decision making in the TBM: the necessity of the pignistic transformation,” Int. J. of Approx. Reasoning, vol. 38, pp. 133–147, 2005. [39] D. Dubois, H. Prade, and P. Smets, “A definition of subjective possibility,” Int. J. of Approx. Reasoning, vol. 48, no. 2, pp. 352–364, 2008.

A PPENDIX A. Internal and external conflict: Other measures We consider other transformations of a mass function into a consonant one, and we show with counterexamples that they cannot be used to discriminate total and external conflicts, as they do not satisfy inequalities (16) and (17) nor Property 9. 1) The pignistic discounting: Another particular aspect of the TBM is the Pignistic transform [38]. It is a unary operator, here denoted by m, that maps a given mass function m into a probability distribution m such that, for ωi ∈ Ω: � m (A) m (ωi ) = ∀ωi ∈ Ω (18) |A| ωi ∈A,A⊆Ω

The set IP(m) ⊂ MΩ denotes the set of all mass assignments whose Pignistic transform gives the same m. The subjective possibility transform [39] transforms an initial probability distribution m into a consonant mass assignment belonging to IP(m). This unary operator is denoted � of the resulting consonant by m � and the contour function pl mass assignment is, for any ωi ∈ Ω, � i) = pl(ω

|Ω| �

min(m(ωi ), m(ωj ))

(19)

j=1

We define the Pignistic discounting as the simple operation consisting in first transforming a mass assignment into its pignistic probability and then applying the possibility transform. � The pignistic discounting of m is simply m.

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This discounting converts a mass function into a consonant one that provides the same decision profile as the original one, and which is the least committed (according to the commonality function) mass from IP(m). Thus, it seems an appealing option to discriminate total and external conflicts. Unfortunately, this transformation does not satisfy Property 9, as shows the next example. Example 11: Consider Ω = {ω1 , ω2 , ω3 } and m1 , m2 m1 ({ω1 , ω2 }) = 0.55 m1 ({ω2 }) = 0.1 m1 ({ω2 , ω3 }) = 0.1 m1 ({ω1 , ω3 }) = 0.25 m2 ({ω1 }) = 0.6

m2 (ω2 ) = 0.4.

We have arg maxω∈Ω pl2 (ω) = arg maxω∈Ω pl1 (ω) = ω1 , hence the external conflict should be null according to Prop�1 and erty 9. However, using the pignistic discounting to get m �2 , we get κ2pl (m �1 , m �2 ) = 0.125, a non-null value. m The main reason for this is due to the fact that the order on elements induced by the pignistic probability is not the same than the one of singleton plausibilities (i.e., we can have pl(ωi ) < pl(ωj ) and m(ωj ) < m(ωi )) 2) The outer consonant approximation: The outer consonant approximation is computed as follows: • order the singletons according to the order induced by the contour function values, i.e., ω1 ≤ ω2 ≤ . . . ≤ ωn with pl(ωi ) ≤ pl(ωj ) if i < j build the possibility distribution (or contour function) pl� such that pl� (ωi ) = P l(Ai ), where Ai = (ω1 , . . . , ωi ). This transformation does satisfy Property 9 but does not always satisfy inequality(17), as shows the next example. Example 12: Consider the same mass functions as in Example 1:

Then T C(m1 , . . . , mN ) and EC(m1 , . . . , mN ) become T C(m1 , . . . , mN ) = 1 − max min {pli (ω)} ω∈Ω i=1,...,N

EC(m1 , . . . , mN ) = 1 − max min {pli� (ω)} ω∈Ω i=1,...,N

while IC(mi ) = 1 − αi . Proving that T C(m1 , . . . , mN ) ≥ maxi=1,...,N {EC(m1 , . . . , mN ), IC(mi )} can be done by using the same arguments as in the binary case. Proving the second inequality T C(m1 , . . . , mN ) ≤ EC(m1 , . . . , mN ) +

1 − plk (ω) ≤ or that

N � i=1

(1 − αi ) + 1 −

pl� (ω) α�

pl� (ω) + α�

�

1−plk (ω) ≤ 1−αk +1−α� +1−

(1−αi ).

i∈{1,...,N } i�=�,k

That 1 − plk (ω) ≤ 1 − αk + 1 − α� + 1 − pl� (ω)/α� can be proved by using the proof for the binary case, and (1 − αi ) for any i ∈ {1, . . . , N } is positive, therefore the inequality remains true.

m1 ({ω1 , ω3 }) = 0.4

m2 ({ω2 , ω3 }) = 0.5

m2 (Ω) = 0.5.

m2 is not changed by the transform, as it is already consonant. m1 becomes m ˜ 1 , such that: m ˜ 1 ({ω1 , ω2 }) = 0.6

IC(mi ) (20)

i=1

goes as follows: as before, taking any ω ∈ Ω, we have that T C(m1 , . . . , mN ) = 1 − maxω∈Ω mini=1,...,N {pli (ω)} ≤ 1 − mini=1,...,N {pli (ω)} As before, we can consider the ω for which the value of EC(m1 , . . . , mN ) is obtained. For this ω, let k be the index such that mini=1,...,N {pli (ω)} = plk (ω), and � the index such that mini=1,...,N {pli� (ω)} = pl� (ω)/α� . Proving Eq. (20) now comes down to show

•

m1 ({ω1 , ω2 }) = 0.6

N �

m ˜ 1 ({ω1 , ω2 , ω3 }) = 0.4

After the combinations, we have T C(m1 , m2 ) = 0.4, IC(m1 ) = IC(m2 ) = 0 and κ2pl (m ˜ 1 , m2 ) = 0. Inequality (17) is not respected. This is due to the conservativeness of the transform that provides very cautious approximations. B. Internal and external conflict: extension to N masses In most cases, the generalization of our results to N mass functions is straightforward. However, the generalization of Proposition 6 is not. In this appendix, we prove that Proposition 6 holds whatever the number of mass functions involved. Let m1 , . . . , mN , αi = maxω∈Ω pli (ω) and pli� = pli/αi denote the mass functions, the maximal values of their contour functions and their transformed plausibilities, respectively.

bilistic models.

Sebastien Destercke received his Ph.D. degree in computer science from Université Paul Sabatier, in Toulouse (France) in 2008. From 2008 to 2011, he was a research engineer at Centre de coopération internationale en recherche agronomique pour le développement.He is currently a researcher with the French National Centre for Scientific Research, in the joint unit Heuristic and Diagnosis for Complex System. His main research interests are in the field of uncertainty modeling and treatment (learning, propagation, information fusion) with imprecise proba-

Thomas Burger received his PhD degree in computer science from Grenoble Institute of Technology, in 2007, while working at Orange Labs. From 2008 to 2011, he was an associate professor at Université de Bretagne Sud. He is currently a researcher with the French National Centre for Scientific Research in the joint unit Institut de Recherche en Technologies et Science pour le Vivant. His research works focus on uncertainty modelling with belief functions, machine learning and high-throughput proteomics.