Selecting Source Behavior in Information Fusion on the Basis of Consistency and Specificity Fr´ed´eric Pichon1 , S´ebastien Destercke2 , and Thomas Burger3 1

2

Thales Research and Technology, Campus Polytechnique, 1 avenue Augustin Fresnel, 91767 Palaiseau cedex, France [email protected] CNRS, UMR Heudiasyc, Centre de Recherches de Royallieu, Compi`egne, France [email protected] 3 CNRS, iRTSV (FR3425), CEA / iRTSV / BGE, INSERM (U1038), Universit´e de Grenoble, France [email protected]

Abstract. Combining pieces of information provided by several sources without prior knowledge about the behavior of the sources is an old yet still important and rather open problem in belief function theory. In this paper, we propose a general approach to select the behavior of sources, based on two cornerstones of information fusion that are the notions of specificity and consistency. This approach is framed in a recently introduced and general fusion scheme that allows a wide range of assumptions on the sources. In the process, we are also led to generalize a recently introduced measure of conflict to all Boolean connectives. Eventually, we show that our approach generalizes some important existing information fusion strategies. Keywords: Dempster-Shafer theory, Information fusion, Consistency, Specificity, Conflict.

1

Introduction

Determining the actual value taken by a variable of interest from information provided by several sources is a central problem in many information systems and has received much attention in belief function theory [16,19]. As argued in [18,14,3], such a task involves necessarily to make some (possibly uncertain) assumptions about the dependence and the behavior, e.g., the relevance and truthfulness [14], of the sources of information. A main concern in information fusion is thus to find what assumption to make about the sources. In this paper, we focus on the problem of finding appropriate source behaviors and assume sources to be independent. When some training data are available, one may resort to some learning procedures to estimate the behavior of the sources (see, e.g., [7,11,6]). When there is no previous experience with the sources (the case in the present paper), then the selection of an appropriate assumption about source behaviors needs to be based on other considerations. L.C. van der Gaag (Ed.): ECSQARU 2013, LNAI 7958, pp. 473–484, 2013. c Springer-Verlag Berlin Heidelberg 2013 

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A first interesting criterion for that choice is the consistency of the knowledge induced on the variable of interest by a given assumption, as suggested by the large body of literature on conflict management (see, e.g., [18,9]). Indeed, it is common in the theory of belief functions to question the behavioral assumptions of the (unnormalized) Dempster’s rule [1,16], i.e., that the sources are truthful and relevant [14], when the conflict or inconsistency [2] resulting from its application is too high. A second natural criterion is the specificity of the induced knowledge. Indeed, there exist assumptions on the sources that, despite their ensuring consistency, are not so often made because they yield poorly informative conclusions. This is the case for instance of the assumption of truthful sources, of which at least one is relevant (the disjunctive rule [4,17] corresponds to this assumption [14]). There might be other relevant criteria to compare assumptions on sources, such as considering a kind of minimal change principle (see, for example [10]) where an assumption could be chosen on the basis of the closeness (in the sense of some distance [8]) of the induced knowledge with respect to the knowledge induced by some reference assumption (e.g., truthful and relevant). In this paper, we propose an approach to select the behavior of sources based on the notions of specificity and consistency (as they are the most classical goals to be reached by a fusion process). This approach is framed in the scheme of Pichon et al. [14], a very general fusion framework that allows making a wide range of assumptions on the sources. In the process, we are led to extend some results presented by Destercke and Burger [2] on conflict measurement. We also show that our approach generalizes some important existing information fusion strategies. We follow a step-wise presentation, first expressing the notions of consistency and specificity in Pichon et al. framework in the case of a single source (Section 3), and then in the case of multiple sources (Section 4). We then describe our approach, and provide some important examples of its application (Section 5). Background material is presented in Section 2. Due to space limitation, proofs are omitted.

2

Preliminaries

In this section, we provide first the necessary concepts about belief function theory and then we recall the formal setting of Pichon et al. [14]. 2.1

Necessary Concepts of Belief Function Theory

In this paper, we assume the beliefs held by an agent about the actual value taken by a given variable x defined on a finite domain X , to be modeled using belief functions [16,19] and to be represented using associated mass functions. Formally, a mass mX on X is a probability distribution on the power  function X X X set 2 , hence A⊆X m (A) = 1. The probability allocation m (A) may be understood as the weight given to the assumption that the agent knows that the value of the variable of interest lies somewhere in set A, and nothing more

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specific [5], or as the probability that the agent supplies information item x ∈ A [14]. Each A ⊆ X such that mX (A) > 0 is called a focal set of the mass function. F denotes the set of focal sets of mX . From the mass function are usually defined two uncertainty measures, the belief and plausibility measures, which respectively reads for an event A ⊆ X :   mX (B) and P l(A) = mX (B). Bel(A) = ∅=B⊆A

B∩A=∅

That is, Bel is the sum of masses of sets that implies A, P l the sum of masses of sets that are consistent with A. The contour function [16] plX : X → [0, 1] associated to a mass function mX is defined by plX (x) = P lX ({x}). There exist several ways to compare the informational contents of belief functions (see, e.g., [5]). In particular, the specialization ordering (the most natural extension of set inclusion) compares belief functions in terms of specificity: mX 1 X X X is a specialization of mX 2 , which we denote by m1  m2 , if and only if m1 can X be obtained from mX 2 by transferring each mass m2 (A) to subsets of A. Many combination rules have been proposed for belief functions [18]: the most common is the unnormalized Dempster’s rule (or conjunctive rule), denoted by X X X ∩ . The mass function m1  ∩ 2 resulting from its application on m1 and m2 is:  X mX mX ∀A ⊆ X . (1) ∩ 2 (A) = 1 1 (B) m2 (C) , B∩C=A

∪ [4,17] is obtained by simply replacing ∩ with ∪ in (1). The disjunctive rule 

2.2

Source Behavioral States

The setting considered by Pichon et al. [14] is the following. Assume an agent wants to know the actual value taken by x based on testimonies provided by several sources of information identified as si , 1 ≤ i ≤ K. These testimonies can be of several forms: a value xi ∈ X , a set Ai ∈ X , a probability distribution pi on X , or in the most general form a mass function mX i on X . In order to be able to interpret those testimonies, the agent must have some knowledge about the behavioral state (referred to as meta-knowledge in [14]) of the sources. In the approach of Pichon et al., the possible elementary behavioral states of a source si are formalized as a set Hi = {hi1 , . . . , hiN }. The set of elementary i joint states on sources is therefore the Cartesian product H1:K := ×K i=1 H . i The state space H can be very general [14] and may include being unreliable, lying, being approximatively informed, etc. Two common assumptions for which we will use specific notations are the assumptions that a source si is relevant (Ri ) or not (¬Ri ), and truthful (T i ) or not (¬T i ). Together, they form the space of possible states Hi = {(T i , Ri ), (T i , ¬Ri ), (¬T i , Ri ), (¬T i , ¬Ri )}. Like the testimonies provided by the sources, the meta-knowledge of the agent can be of several form, the most general one being a mass function defined over H1:K . In the following, we detail how consistency and specificity can be characterized when using this setting, and how such characterizations can be used to select a particular piece of meta-knowledge.

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Consistency and Specificity: Single Source

We start by characterizing consistency and specificity in the simple case where a single source provides information. 3.1

Crisp Testimony and Sure Meta-knowledge

The simplest situation is a source s delivering a testimony of the form x ∈ A with A ⊆ X , and being known to be in a state h ∈ H, with H the state space of the source. The testimony x ∈ A should then be modified according to this state [14]. This transformation can be encoded by a multivalued mapping ΓA : H → X , where ΓA (h) indicates how to interpret the piece of information x ∈ A for each possible state h of the source. For instance, if H = {(T, R), (T, ¬R), (¬T, R), (¬T, ¬R)} are the possible states of the source, we have for all A ⊆ X ΓA (R, T ) = A, ΓA (¬R, T ) = X , ΓA (R, ¬T ) = Ac , ΓA (¬R, ¬T ) = X ,

(2)

with Ac the complement of A. Eqs. (2) translate that if s is considered not relevant, it does not bring any information, while if it is considered not truthful, it declares the opposite of what it knows to be true – this corresponds to the crudest form of non-truthfulness, other forms are discussed in Pichon [12]. If the knowledge about the source state is imprecise and given by H ⊆ H, then the  transformation is the image ΓA (H) := h∈H ΓA (h) of H by ΓA . Destercke and Burger [2] consider that any piece of knowledge x ∈ A about a variable x is consistent if A = ∅, and inconsistent otherwise. This extends easily to the current framework, a transformed testimony yielding a consistent piece of knowledge on X when ΓA (H) = ∅, in which case x ∈ A is said H-consistent, and an inconsistent piece of knowledge when ΓA (H) = ∅. We may then adapt the measure of consistency introduced in [2] to measure H-consistency as the degree φH : 2X → {0, 1} such that  1 if ΓA (H) = ∅, φH (A) = 0 if ΓA (H) = ∅. In some way, this consistency measure evaluates whether H is a valid assumption on the source when it provides the testimony x ∈ A. Consider, for instance, the assumption h = (R, ¬T ) corresponding to a relevant and lying source. This assumption will be considered invalid only when the source provides the testimony x ∈ X as ΓX (h) = ∅ and φh (X ) = 0. Meta-knowledge can also be characterized in terms of specificity: namely a piece of meta-knowledge H1 ⊆ H will be said at least as meta-specific as another piece of meta-knowledge H2 ⊆ H when ΓA (H1 ) ⊆ ΓA (H2 ) for any A ⊆ X , and we will denote it H1 H H2 . For example, the assumption (R, T ) is at least as meta-specific as the assumption (¬R, T ). Note that we have the relations H1  H2 ⇒ H1 H H2 and H1 H H2 ⇒ φH1 (A) ≥ φH2 (A), the latter relation being of particular interest in the context of this paper as it shows that reaching both consistency and specificity are somewhat opposite goals.

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477

Uncertain Testimony and Meta-knowledge

More generally, both the testimony and the meta-knowledge of the agent may be uncertain. Let mX be the uncertain testimony and mH the uncertain metaknowledge. The knowledge of the agent on X can then be represented by the mass function m[mH ]X defined for all B ⊆ X as [14]   mH (H) mX (A). (3) m[mH ]X (B) = H⊆H

A:ΓA (H)=B

This definition is rather general. In particular, the discounting rule proposed by Shafer [16] is retrieved by mH (R) = p and mH (¬R) = 1 − p [14]. The results of the previous section can be extended to this general setting: following [2], the mass function modeling the empty set (m[mH ]X (∅) = 1) can be associated to a complete inconsistent knowledge and a mass function m[mH ]X whose focal sets have a non-empty intersection can be associated to a totally consistent knowledge. That is, the testimony mX is totally consistent under meta-knowledge mH if and only if  ΓA (H) = ∅, (4) A∈F H∈FH

where F and FH denote the sets of focal sets of mX and mH , respectively. A mass function mX is then said mH -consistent if and only if (4) holds. Lemma 1 characterizes mH -consistent testimonies in terms of the contour function.  H X Lemma 1. A∈F ΓA (H) = ∅ ⇔ ∃x ∈ X such that pl[m ] (x) = 1, where H∈FH

pl[mH ]X is the contour function associated to the mass function m[mH ]X obtained from (3). A source is thus mH -consistent if it allows us to conclude that at least one value of x is totally plausible under meta-knowledge mH . Following [2], this characterization of mH -consistency suggests the following definition: Definition 1 (mH -consistency measure). The measure φmH : MX → [0, 1] of mH -consistency, where MX denotes the set of all mass functions on X , reads: φmH (mX ) = max pl[mH ]X (x). x∈X

The notion of meta-specificity may also be extended to this general setting. Definition 2 (Meta-specificity). An uncertain piece of meta-knowledge mH 1 is said to be at least as meta-specific as another uncertain piece mH 2 when X H X H m[mH for any mX ∈ MX . This is denoted by mH 1 ]  m[m2 ] 1  H m2 . We may then show that in the general case, consistency and specificity are also at odds: H Proposition 1. If mH (mX ) ≤ φmH (mX ) ∀ mX ∈ MX . 1 H m2 , then φmH 1 2

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Example 1 (Inspired from Example 1 of [14]). Let X = {x1 , x2 , x3 , x4 , x5 } be an ordered space and consider the mass function such that mX ({x1 , x2 }) = 0.3, mX ({x4 , x5 }) = 0.3 and mX ({x3 }) = 0.4. Now consider the assumptions h1 “informed” such that ΓA (h1 ) = A, h2 “approximately informed” such that if A = {xi , xi+1 , . . . , xj } then ΓA (h2 ) = {xi−1 } ∪ A ∪ {xj+1 } with x0 = x6 = ∅, and h3 “unreliable” such that ΓA (h3 ) = X . Then we have φh1 (mX ) = 0.4,

φh2 (mX ) = 1,

φh3 (mX ) = 1,

h1  H h2  H h3 . This example allows us to lay bare some preliminary ideas on the selection of source behavior based on consistency and specificity. As can be seen, assumptions h2 and h3 are the most desirable in terms of consistency, since they both yield a totally consistent state of knowledge on X . However, the state of knowledge obtained under h2 is more specific, or informative, than the one obtained under h3 , hence h2 may appear preferable. Those ideas will be developed at length in Section 5.

4

Consistency and Specificity: Multiple Sources

We now  consider multiple sources si , i = 1, . . . , K where each can be in states Hi = hi1 , ..., hiN and deliver testimonies mX i , i = 1, . . . , K. We define for any state h = (h1 , . . . , hk ) ∈ H1:K a mapping [14] for any A = (A1 , . . . , AK ) ⊆ X K K as ΓA (h) = i=1 ΓAi (hi ). ΓA (h) is the information on X deduced from testimonies (A1 , . . . , AK ) of sources s1 , . . . , sK when they are in states (h1 , . . . , hK ). We keep the notation ΓA (H) := ∪h∈H ΓA (h) for all H ⊆ H1:K and all A ⊆ X K . 4.1

General Case 1:K

over ×K If we have a joint meta-knowledge mH i=1 Hi and if sources s1 , . . . , sK 1:K are independent, then the combined mass function m[mH ]X defined by (5) X represents what can be inferred about x from mX = (mX 1 , ..., mK ) [14]:

K    H1:K X H1:K X ] (B) = m (H) mi (Ai ) . (5) m[m H⊆H1:K

A⊆X K ΓA (H)=B

i=1

We note that this approach has a computational complexity that increases exponentially in the number of sources. Keeping the same definition of complete inconsistent and consistent knowledge as in Section 3.2, the counterpart of Lemma 1 suggests to use the following 1:K equation as a degree of mH -consistency for the collection mX φmH1:K (mX ) = max pl[mH x∈X

1:K

]X (x),

(6)

Selecting Source Behavior in Information Fusion 1:K

1:K

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1:K

where pl[mH ] is the contour function of (5). Again, if mH  H mH , then 1 2 X X X φmH1:K (m ) ≤ φmH1:K (m ) for any m . We will make heavy use of this duality 1 2 between specificity and consistency in Section 5. An interesting feature of the approach [14] is that all Boolean operators on sets A = (A1 , . . . , AK ) ⊆ X K can be obtained through particular assumptions on the behavior of the sources. As a result, Equation (5) covers all combination rules based on Boolean operators. For instance, consider the assumption HrK on H1:K meaning the sources are truthful and “r-out-of-K” of them are relevant. This amounts to

(∩A∈A A) , (7) ΓA (HrK ) = A⊂{A1 ,...,AK },|A|=r ∩ and and when applying HrK to Eq. (5), the conjunctive and disjunctive rules  ∪ are retrieved when r = K and r = 1, respectively. 

Remark 1 (Extension of conflict to all Boolean operators). This feature, once coupled with Equation (6), is fruitful: it provides a natural extension of the measure of conflict defined in [2] as the inconsistency resulting from the conjunctive combination, to all other combination rules based on Boolean operators. 4.2

Separable Meta-knowledge

Computing (5) can be resource demanding, however there are cases where it is 1:K are separable. easier. In particular, when all focal elements of mH Definition 3 (Separability). A subset H ⊆ H1:K is said separable if and only if H = H ↓1 × . . . × H ↓K , where H ↓i denotes the projection of H ⊆ H1:K on Hi . 1:K

Proposition 2. When each focal set of mH is separable1 , Equation (5) can be rewritten as:    K 1:K 1:K ∩ i=1 m[H ↓i ]X (B), (8) mH (H) ·  m[mH ]X (B) = H⊆H1:K ↓i where m[H ↓i ]X denotes mass function mX i transformed according to H . ↓i That is, we first transform each mX i according to H , apply unnormalized Demp1:K ster’s rule to them and compute the weighted sum according to mH . We can therefore make use of efficient algorithms to compute Dempster’s rule result [20]. This property also simplifies the computation of the consistency measure (6). 1:K Indeed, consider the meta-knowledge mH (H) = 1 with H separable and let pl[H]X be the corresponding contour function. Then if pl[H ↓i ]X is the contour 1

1:K

This happens, e.g., when mH satisfies the property of meta-independence [14], 1:K is the result of independent pieces of metawhich basically means that mH knowledge concerning each source.

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↓i function obtained by transforming mX i according to meta-knowledge H , we K X ↓i X have pl[H] (x) = i=1 pl[H ] (x). As Equation (8) is a convex mixture of such mass functions, and as the plausibility measure of a convex mixture is the convex mixture of plausibility measures, computing consistency measure (6) only requires to compute contour functions and to take their weighted averaged products (hence not necessitating any combination).

5

Source Behavior Selection Approach

Selecting which assumption to make on the sources when one has no previous experience with them, basically amounts to defining a set of candidate pieces of meta-knowledge, and a selection criterion allowing one to choose a particular element in this set. Based on the results of the previous sections, this section provides some guidelines to define such a set, as well as a selection criterion that can be used on any set satisfying those guidelines, leading to a general, yet practical and sensible, approach to select the behavior of the sources. Important examples of the application of this approach are also presented. 5.1

Initial Meta-knowledge

In absence of any particular information on the behavior of the sources, we 1:K 1:K such that mH (h) = 1, with propose to consider first an assumption mH 1 1 1:K ↓i h ∈ H and ΓA (h ) = A, ∀A ⊆ X , i = 1, ..., K, i.e., an assumption that induces no transformation of the testimonies provided by the sources. This assumption corresponds to an agent that does not want to alter in any way the information he has received: it amounts to accepting the testimonies as they are. Most importantly, the assumption that the sources are all relevant and truthful, i.e., the most classical assumption in information fusion in general and in be1:K . Our proposal lief function theory in particular, is formally an instance of mH 1 corresponds indeed to combining the sources using the unnormalized Dempster’s rule – the first rule usually considered to combine pieces of information. Hence, 1:K is a natural default meta-knowledge. mH 1 1:K Equation (6) provides us with an assessment of whether the assumption mH 1 applies to the current testimonies. In particular, and as is classically advocated in belief function theory, we propose that if the consistency induced by this assumption is high enough, that is if it is above some threshold τ , then this assumption should be used to combine the testimonies, and if the consistency is 1:K should not be used and other too low, i.e., below τ , then the assumption mH 1 assumptions leading to higher consistency should be sought. 5.2

A Specificity Ordering Approach

To search for other assumptions with better consistency, the counterpart of Proposition 1 in the multiple source case can be instrumental: choosing a

Selecting Source Behavior in Information Fusion 1:K

1:K

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1:K

meta-knowledge mH such that mH  H mH will indeed ensure that the 2 1 2 consistency increases. This leads us to propose the following strategy to select the meta-knowledge to be used: 1:K

1:K

1:K

= (mH , ..., mH ) such that – define a collection of meta-knowledge mH 1 M H1:K H1:K H1:K H mj+1 , and with m1 as defined above; for any 1 ≤ j < M , mj H1:K – test each mj iteratively with j = 1, . . . , M , until φmH1:K (mX ) ≥ τ . j

In other words, this strategy gradually decreases specificity until a satisfactory consistency level is reached. It comes down to considering a set of pieces of meta1:K being the most knowledge that are comparable according to H , with mH 1 meta-specific element of this set, and to select in this set the most meta-specific 1:K such that φmH1:K (mX ) ≥ τ . element mH j j

1:K

Remark 2. The construction of mH should also follow some sensible rules: 1:K should have a clear semantic and the spaces Hi pieces of meta-knowledge mH j should be of reduced size, e.g., Hi = {(T i , Ri ), (T i , ¬Ri ), (¬T i , Ri ), (¬T i , ¬Ri )}. 5.3

Examples

As shown below, our approach subsumes important classical fusion strategies dedicated to conflict management in belief function theory. These strategies follow the same pattern: they first combine the testimonies using the unnormalized Dempster’s rule, and if the consistency resulting from its application is too low, other assumptions on the sources yielding higher consistency are considered. Let us remark that the first fusion strategy discussed below is based on imprecise pieces of meta-knowledge, whereas the second one is based on probabilistic ones. r-out-of-K Relevant Sources. We can implement the above methodology by 1:K K K choosing mH (HK−j+1 ) = 1, with HK−j+1 the assumption that the sources j are truthful and r = K − j + 1 out of them are relevant (see Eq. (7)), as the following proposition indicates: Proposition 3. If mH j

1:K

K (HK−j+1 ) = 1, then mH j

1:K

1:K

 H mH j+1 for 1 ≤ j < K.

X X Example 2. Consider the mass functions mX 1 , m2 and m3 on X = {x1 , x2 , x3 } in the left part of Table 1. Assume they were received from three independent 1:K 1:K 1:K 1:K = (mH , mH , mH ) = (H33 , H23 , H13 ) be three pieces of sources. Let mH 1 2 3 1:K corresponds to the use meta-knowledge we want to test on these sources. mH 1 1:K H of the unnormalized Dempster’s rule, while m3 corresponds to the use of the 1:K corresponds to the assumption H23 that the three sources disjunctive rule. mH 2 are truthful and that two of them are relevant, but we do not know which ones, i.e., to the following subset of H1:K :

{(R1 , T1 , R2 , T2 , ¬R3 , T3 ) , (R1 , T1 , ¬R2 , T2 , R3 , T3 ) , (¬R1 , T1 , R2 , T2 , R3 , T3 )} . (9)

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F. Pichon, S. Destercke, and T. Burger Table 1. Mass functions resulting from the three different assumptions A ∅ {x1 } {x2 } {x1 , x2 } {x3 } {x1 , x3 } {x2 , x3 } X

X X 3 X mX m[H23 ]X m[H33 ]X 1 m2 m3 m[H1 ] 0 0 0 0 0 0.36 0.5 0 0 0 0.06 0.2 0 0 0 0 0 0 0 0.2 0 0 0.04 0.04 0 0 0.6 0 0 0.24 0 0 0 0 0.24 0 0 0 0 0 0 0 1 0.66 0.16 0.5 0.8 0.4

The right part of Table 1 presents the mass functions on X resulting from the three different assumptions. We have φH13 (mX ) = 1, φH23 (mX ) = 1 and φH33 (mX ) = 0.4, hence our approach suggests to use H23 to combine the pieces of information in this example. Note that the assumption “r-out-of-K” is not separable in general. For instance, the subset (9) is not the product of each of its projection. However, we may remark that this assumption treats all sources in the same way, which seems interesting in absence of meta-knowledge about each individual source. Vectors of Reliabilities. Another interesting case is when we consider Hi = i {Ri , ¬Ri } (relevant or not) and a vector p = (p1 , . . . , pK ) such that mH (Ri ) = i 1:K pi , mH (¬Ri ) = 1−pi and where mH is obtained by considering the stochastic 1:K amounts product of probabilities p1 , . . . , pk . In such case, the assumption mH to discounting each source si according to reliability rate 1 − pi and then combining the discounted sources using unnormalized Dempster’s rule [14]. If we define a set p1 , . . . , pM of such vectors with pji > pj+1 , we get corresponding i H1:K H1:K , . . . , mM with the following property. meta-knowledges m1 Proposition 4. Let mH j M

p , . . . , p . We have 1

1:K

, j = 1, . . . , M, be the mass functions defined using

1:K mH j

1:K

 H mH j+1 , for 1 ≤ j < M .

1:K

1:K

A useful feature of such mH is that each meta-knowledge mH satisfies the j X meta-independence property [14] and therefore φmH1:K (m ) can be computed j

efficiently using the results of Section 4.2. Remark 3. If we associate pji with the product of one minus the degrees of falsity of mass function i up to step j in Schubert’s recent work on sequential discount1:K ]X is nothing else but the mass function on X obtained ing [15], then m[mH j at step j in Schubert’s scheme. Hence Schubert’s method [15] is included in the present approach.

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Conclusion

In this paper, we have proposed a practical and sensible methodology to select the behavior of sources in information fusion, based on the fundamental notions of specificity and consistency. Our approach is based on recent frameworks that measure inconsistency [2] and model source behaviors [14] in simple yet powerful ways. In particular, we have introduced measures of consistency and a partial ordering for the source behavior assumptions allowed by Pichon et al. framework [14], which are used in the behavior selection process. This also led notably to a natural extension of the measure of conflict defined in [2], to all combination rules based on Boolean operators. In addition, an interesting feature of our approach is that it subsumes important classical fusion strategies dedicated to conflict management in belief function theory. We may mention a few research paths that were left unexplored in this paper: – as in [2], it would be interesting to study the alternative consistency measure based on mX (∅), or what happens in the current framework when we relax the assumption of source independence; – variations of our approach could be investigated, both from formal and practical point of views, and in particular using other criteria than specificity and consistency, for instance the idea of minimal change evoked in Section 1; – besides the families of assumptions on the sources that are studied in Section 5.3, it may be interesting to identify other families of assumptions that are ordered according to the relation of meta-specificity and that include the unnormalized Dempster’s rule as most meta-specific element; – if several collections mX of testimonies are available, then one could try to learn the best meta-knowledge to be used in general to combine the testimonies. In particular, we may think of obtaining a probability distribution 1:K and exploit it for selecting the best meta-knowledge. This inforover mH mation could be coupled with other methods that learns reliability indices [7,11,6]. – one could try to integrate in the current framework some related works, such as Smets expert system [18] or Mercier et al. [11] contextual discounting; – the idea of using consistency and specificity as rule selection methods could be extended to rules that have no clear interpretation in terms of metaknowledge, such as weight-based ones [13]. Acknowledgements. This work was partially carried out in the framework of (1) the ANR funding ANR-11-IDEX-0004-02 (Labex MS2T, “Investissements d’Avenir” call) (2) the ANR funding ANR-10-INBS-08 (ProFI project, “Infrastructures Nationales en Biologie et Sant´e”; “Investissements d’Avenir” call) , and (3) the Prospectom project of the Mastodons 2012 challenge (CNRS).

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