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On the Conflict Measures Agreed with the Combining Rules Alexander Lepskiy National Research University Higher School of Economics, Moscow, Russia BELIEF 2018, September 17 - 21, 2018, Compi` egne, France A. Lepskiy (HSE) Coherence of

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On the Conflict Measures Agreed with the Combining Rules

Alexander Lepskiy

National Research University – Higher School of Economics, Moscow, Russia BELIEF 2018, September 17 - 21, 2018, Compi` egne, France

  • A. Lepskiy

(HSE) Coherence of the Conflict BELIEF 2018 1 / 21

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Preamble

Research Motivation

The value of a conflict measure (CM) between bodies of evidence (BE) is the important characteristic for deciding about applying of combining rule. The main directions of research on the conflict measure: axiomatics of CM between the BE (external conflict) [Martin 2012,

Destercke & Burger 2013, Bronevich et al. 2015] ;

different approaches to evaluation of conflict between the BE

[Jousselme et al. 2001, Liu 2006, Jousselme & Maupin 2012, Martin 2012, Lepskiy 2013, ...] ;

internal conflict of evidence [Daniel 2014, Schubert 2012, Lepskiy 2016,

...] ;

conflict management [Lefevre et al. 2001, Schubert 2011, Martin 2018, ...] ; ... etc.

  • A. Lepskiy

(HSE) Coherence of the Conflict BELIEF 2018 2 / 21

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Preamble

Research Motivation

Main thesis The choice of a specific measure for estimation of a conflict depends on a solvable problem. For example, if we estimate conflict between BE with the aim of decision making about combining of evidence, then the conflict measure must be agreed with the combining rule. The main aims of this research are to study the CMs that are induced by conjunctive and disjunctive combining rules; the relationship between the conditions of consistency and axioms

  • f CM;

the structure of conjunctive and disjunctive CMs.

  • A. Lepskiy

(HSE) Coherence of the Conflict BELIEF 2018 3 / 21

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Preamble

Outline of Presentation

the background of evidence theory; axioms of CMs; CMs induced by conjunctive and disjunctive combining rules; the coherence of CMs and combining rules; metric and entropic components of a CM; conclusions.

  • A. Lepskiy

(HSE) Coherence of the Conflict BELIEF 2018 4 / 21

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Evidence Theory and Combining Rules

Background of Evidence Theory

X be a some universal set of all possibility values of experimental results, P(X) be a powerset of X; a mass function m : P(X) → [0, 1],

A∈P(X) m(A) = 1;

A ⊆ X is called a focal element if m(A) > 0; A = {Ai} be a set of all focal elements of evidence; the pair F = (A, m) is called a body of evidence, BE; F(X) be a set of all BE on X. BE is said to be categorical (is denoted as FA = (A, 1)) if it has

  • nly one focal element; BE FX = (X, 1) is said to be vacuous;

if Fj = (Aj, mj) ∈ F(X) and

j αj = 1, αj ∈ [0, 1] ∀j, then

F =

j αjFj = (A, m) ∈ F(X), where A = j Aj,

m(A) =

j αjmj(A); we have F = A∈A m(A)FA ∀F = (A, m).

  • A. Lepskiy

(HSE) Coherence of the Conflict BELIEF 2018 5 / 21

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Evidence Theory and Combining Rules

Conjunctive Combining Rules

Let we have two BE F1 = (A1, m1) and F2 = (A2, m2) which represent two information sources. The different combining rules R are considered in evidence theory: R : F(X) × F(X) → F(X). For example, the non-normalized conjunctive rule D0(F1, F2) is considered mD0(A) =

  • B∩C=A m1(B)m2(C),

A ∈ 2X. The conjunctive conflict measure KD(F1, F2) = mD0(∅) characterizes the amount of conflict between two sources of information described by the F1 and F2. If KD = 1, then the classical Dempster rule for combining of two BE can be defined: mD(A) = 1 1 − KD mD0(A), A = ∅, mD(∅) = 0.

  • A. Lepskiy

(HSE) Coherence of the Conflict BELIEF 2018 6 / 21

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Evidence Theory and Combining Rules

Disjunctive Combining Rule

Dubois and Prade’s disjunctive consensus rule is a dual rule to Dempster’s rule in some sense: mDP (A) =

  • B∪C=A m1(B)m2(C),

A ∈ 2X. The duality relation is true by analogy with De Morgan’s law: D0(F1, F2) = DP

  • F1, F2
  • ,

where the complement BE F =

A∈A m(A)FA, if F = A∈A m(A)FA.

  • A. Lepskiy

(HSE) Coherence of the Conflict BELIEF 2018 7 / 21

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Axioms of Conflict Measures

Axioms of Conflict Measures

In general, it is desirable that the CM K(F1, F2) between BE satisfies the following conditions (axioms) [Martin 2012, Destercke & Burger 2013,

Bronevich et al. 2015, ...] :

A1: 0 ≤ K(F1, F2) ≤ 1 ∀F1, F2 ∈ F(X) (non-negativity and normalization); A2: K(F1, F2) = K(F2, F1) ∀F1, F2 ∈ F(X) (symmetry); A3: K(F, F) = 0 ∀F ∈ F(X) (nilpotency); A4: K(F ′, F) ≥ K(F ′′, F), if F ′ = (A′, m), F ′′ = (A′′, m), where A′ = {A′

i}, A′′ = {A′′ i } and A′ i ⊆ A′′ i ∀i (antimonotonicity with

respect to imprecision of evidence); A5: K(FX, F) = 0 ∀F ∈ F(X) (ignorance is bliss); A6: K(FA, FB) = 1, if A ∩ B = ∅.

  • A. Lepskiy

(HSE) Coherence of the Conflict BELIEF 2018 8 / 21

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CMs Induced by Combining Rules

CM Induced by Conjunctive Rule

KD(F1, F2) =

  • B∩C=∅ m1(B)m2(C).

This CM satisfies all axioms with exception A3. Special case. Let we have disjoint belief structure A, i.e. A′ ∩ A′′ = ∅ ∀A′, A′′ ∈ A (A′ = A′′). Then KD(F1, F2) = 1 −

  • B m1(B)m2(B).

In this case the measure KD is close to KH(F1, F2) = 1 −

  • B
  • m1(B)m2(B) =
  • 1

√ 2 √m1 − √m2

2 . The

1 √ 2

  • √m1 − √m2
  • is Hellinger distance between probability

distributions m1 and m2. The CM KD is close also to KM(F1, F2) = 1 −

  • B min {m1(B), m2(B)}

[Bronevich & Rozenberg 2016] on the disjoint belief structure.

  • A. Lepskiy

(HSE) Coherence of the Conflict BELIEF 2018 9 / 21

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CMs Induced by Combining Rules

CM Induced by Disjunctive Consensus Rule

KDP (F1, F2) = mDP (X) =

  • B∪C=X m1(B)m2(C).

Proposition

1 KDP (F1, F2) = KD(F1, F2); 2 if F1 = (A1, m1), F2 = (A2, m2) and A1 = ¯

A2 = { ¯ A : A ∈ A2}, then KD(F1, F2) = 1 ⇒ KDP (F1, F2) = 1. Note that the measure 1 − KDP satisfies almost all axioms (A1, A2, A4, A5), but this measure counter-intuitive with A6.

  • A. Lepskiy

(HSE) Coherence of the Conflict BELIEF 2018 10 / 21

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CMs Induced by Combining Rules

The disjunctive CM satisfies the following conditions: A1: 0 ≤ KDP (F1, F2) ≤ 1 ∀F1, F2 ∈ F(X) (non-negativity and normalization); A2: KDP (F1, F2) = KDP (F2, F1) ∀F1, F2 ∈ F(X) (symmetry); A4′: KDP (F ′, F) ≤ KDP (F ′′, F), if F ′ = (A′, m), F ′′ = (A′′, m), where A′ = {A′

i}, A′′ = {A′′ i } and A′ i ⊆ A′′ i ∀i (monotonicity with

respect to imprecision of evidence); A5′: KDP (F∅, F) = 1 ∀F ∈ F(X)\{F∅} (maximum conflict with extraneous evidence). In this case we assume that the empty set can be a focal element; A6′: KDP (FA, FA) = 1.

  • A. Lepskiy

(HSE) Coherence of the Conflict BELIEF 2018 11 / 21

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CMs Induced by Combining Rules

Example

Let F1 = α1F{x1} + α2F{x2} + (1 − α1 − α2)F{x1,x2}, F2 = β1F{x1} + β2F{x2} + (1 − β1 − β2)F{x1,x2} be two BE on X = {x1, x2}. Then KD(F1, F2) = |[α, β−]| , KDP (F1, F2) = 1 − (α, β), where α = (α1, α2), β = (β1, β2), β− = (β1, −β2), [·, ·] is a vector product,(·, ·) is a scalar product. We have KD(F1, F2) ≤ KDP (F1, F2). In particular, KD(F1, F2) = 1 ⇔ α = (1, 0), β = (0, 1) and vice versa; KDP (F1, F2) = 1 ⇔ α⊥β ⇔ α = (α, 0) ∧ β = (0, β) or vice versa. Analogically, KD(F1, F2) = 0 ⇔ α = (α, 0), β = (β, 0) and vice versa; KDP (F1, F2) = 0 ⇔ α = (1, 0) ∧ β = (1, 0) or vice versa.

  • A. Lepskiy

(HSE) Coherence of the Conflict BELIEF 2018 12 / 21

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The Coherence of CM and Combining Rules

The Coherence of CM and Combining Rules

The value of a CM is an important factor for decision making about using of combining rules for aggregation of information. It is clear that the choice of a combining rule and a CM must be coordinated to a certain degree in such problems. Definition A combining rule R and a CM K are called positively agreed if K(F1, F2) ≤ K(R(F1, F2), Fi), i = 1, 2 ∀F1, F2 ∈ F(X). The pair R and K is called negatively agreed if the opposite inequality holds. Note that the negative coherence of the pair R and K is associated with the so-called property of conjunctively consistency for dissimilarity K and combining rule D0, which was considered in

[Loudahi et al. 2014] : K(D0(F1, F3), D0(F2, F3)) ≤ K(F1, F2).

  • A. Lepskiy

(HSE) Coherence of the Conflict BELIEF 2018 13 / 21

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The Coherence of CM and Combining Rules

Proposition

1 the combining rule D0 and the CM KD are positively agreed; 2 the combining rule DP and the CM KDP are positively agreed.

The coherence of a CM with a combining rule makes some axioms dependent or contradictory. For example, if K and D or K and DP are positively agreed, then A3 ⇒ A5; if K and D are positively agreed, then A6 ⇒ K(F∅, FA) = 1 for all A = ∅ (particular case of axiom A5′); if K and DP are positively agreed, then A5 and A6 axioms are contradictory as well as A3 and A5′; if K and D are negatively agreed, then axiom A5 ⇒ A3; if K and D are negatively agreed, then A5′ ⇒ K(FA, F ¯

A) = 1 for

all A = ∅ (particular case of axiom A6).

  • A. Lepskiy

(HSE) Coherence of the Conflict BELIEF 2018 14 / 21

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Metric and Entropic Components of a CM

Metric and Entropic Components of a CM

  • Example. Let us assume that there are three candidates

X = {x1, x2, x3} for a certain position. Three experts expressed their preference to these candidates as three BE F1 = 1

3F{x1} + 1 3F{x2} + 1 3F{x3}, F2 = 1 3F{x1,x2} + 2 3F{x3},

F3 = 7

8F{x2} + 1 8F{x2,x3}.

The CMs are equal KD(F1, F2)= 5

9, KD(F1, F3)= 5 8, KD(F2, F3)= 7 12.

We will choose for combining a couple F1 and F2 with the lowest

  • conflict. We get D(F1, F2) = 1

4F{x1} + 1 4F{x2} + 1 2F{x3}, i.e. the

preference is given to a third candidate. At the same time, the evidence F1 is irrelevant because the first expert did not give preference to any of the candidates. If we find a combination of F2 and F3, then we get D(F2, F3) = 4

5F{x2} + 1 5F{x3},

i.e. the preference is given to the second candidate in this case. The first BE has a uniform probability distribution. It has high Shannon entropy and it is better not to use for combining.

  • A. Lepskiy

(HSE) Coherence of the Conflict BELIEF 2018 15 / 21

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Metric and Entropic Components of a CM

Let xQ :=

  • xT Qx, Q = (qB,C) is a symmetric positive definite

matrix which satisfies the conditions: 1) qB,C ∈ [0, 1] ∀B, C ∈ 2X; 2) qB,C = 0, if B ∩ C = ∅; 3) qB,B = 1 ∀B ∈ 2X. In particular, if qB,C = |B∩C|

|B∪C|, ∀B, C = ∅ is Jaccard index, then

dJ(F1, F2)= 1

√ 2 m1 − m2Q is Jousselme distance [Jousselme et al. 2001] .

Consider the functional EQ : F(X) → [0, 1] EQ(F)=EQ(m)=1 − m2

Q =

  • B m(B)
  • 1 −
  • C qB,Cm(C)
  • .

This functional is close to an entropy functional: t(min)=arg min EQ(t), if ∃j : t(min)

j

=1 and t(min)

k

=0 ∀k=j (categorical BE), EQ( t(min))=0. Let rB,C =

  • 0,

B ∩ C = ∅ ∨ B = C, 1 − qB,C,

  • therwise.
  • A. Lepskiy

(HSE) Coherence of the Conflict BELIEF 2018 16 / 21

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Metric and Entropic Components of a CM

Proposition KD(F1, F2)=1

2

  • EQ(m1)+EQ(m2)+m1−m22

Q

  • −
  • B,C

rB,Cm1(B)m2(C). This formula shows that the conjunctive CM can be represented as a sum of average value of entropy-type functionals of BE, the square distance between BE and a last summand that characterizes the interaction of weakly intersecting focal elements. Corollary If F1 = (A, m1) and F2 = (A, m2), where A′ ∩ A′′ = ∅ ∀A′, A′′ ∈ A, then KD(F1, F2) = 1

2 (EI(m1) + EI(m2)) + 1 2 m1 − m22 I ,

where I is the identity matrix and xI := √ xT x is the Euclidean norm.

  • A. Lepskiy

(HSE) Coherence of the Conflict BELIEF 2018 17 / 21

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Metric and Entropic Components of a CM

Functional EI(t) =

B t(B)(1 − t(B)) satisfies the conditions:

t(max) = arg max EI(t), if t(max)

k

=

1 2|X|−1 ∀k (uniform

distribution); EI(t(max)) = 1 −

1 2|X|−1; t(min) = arg min EI(t), if ∃j : t(min) j

= 1 and t(min)

k

= 0 ∀k = j (categorical evidence), EI(t(min)) = 0; EI(t) ≤ S(t) := −

k tklog2tk (Shannon entropy).

Thus, the conjunctive CM is equal in this case the average value of the entropy-type functionals and the square of the distance between the mass functions of the two BE. The conjunctive CM satisfies the triangle inequality on the disjoint belief structures. Proposition If Fi = (A, mi), i = 1, 2, 3, where A′ ∩ A′′ = ∅ ∀A′, A′′ ∈ A, then KD(F1, F3) ≤ KD(F1, F2) + KD(F2, F3).

  • A. Lepskiy

(HSE) Coherence of the Conflict BELIEF 2018 18 / 21

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Summary and Conclusion

Summary and Conclusion

1 the notion of a disjunctive conflict measure was introduced and

some duality relations between the conjunctive and disjunctive conflict measures are established;

2 some of the consistency conditions between the combining rules

and conflict measures were discussed. The relationship of consistency conditions and the axiomatic of a conflict measure is shown;

3 it is shown that the metric and entropic components can be

isolated into the conjunctive conflict measures. It is shown that the entropic component of evidence is an important characteristic (together with the value of a conflict measure) in decisions about the choice of the bodies of evidence for combining.

  • A. Lepskiy

(HSE) Coherence of the Conflict BELIEF 2018 19 / 21

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References

References

Bronevich, A., Lepskiy, A., Penikas, H.: The application of conflict measure to estimating incoherence of analyst’s forecasts about the cost of shares of Russian companies. Procedia Computer Science. 55, 1113–1122 (2015) Bronevich, A.G., Rozenberg, I.N.: Conjunctive rules in the theory of belief functions and their justification through decisions models. In: Vejnarov´ a, J., Kratochv´ ıl, V. (eds.) BELIEF, LNCS, 9861, 137–145. Springer Verlag, Berlin (2016) Daniel, M.: Conflict between belief functions: a new measure based on their non-conflicting parts. In: Cuzzolin, F. (ed.) BELIEF, LNCS, 8764, 321–330. Springer, Heidelberg (2014) Destercke, S., Burger, T.: Toward an axiomatic definition of conflict between belief functions. IEEE Transactions on Cybernetics. 43(2), 585–596 (2013) Jousselme, A.-L., Grenier, D., Boss´ e, ´ E.: A new distance between two bodies of evidence. Information Fusion. 2, 91–101 (2001) Lepskiy, A.: Decomposition of evidence and internal conflict. Procedia Computer Science. 122, 186–193 (2017) Liu, W.: Analysing the degree of conflict among belief functions. Artificial Intelligence. 170, 909–924 (2006) Loudahi, M., Klein, J., Vannobel, J.-M., Colot, O.: New distances between bodies of evidence based

  • n Dempsterian specialization matrices and their consistency with the conjunctive combination
  • rule. Intern. J. of Approx. Reasoning. 55(5), 1093–1112 (2014)

Martin, A.: About conflict in the theory of belief functions. In: Denœux, T., Masson, M-H. (eds.) BELIEF, AISC, 164, 161–168. Springer, Heidelberg (2012) Schubert, J.: The internal conflict of a belief function. In: Denœux, T., Masson, M-H. (eds.) BELIEF, AISC, 164, 169–177. Springer, Heidelberg (2012)

  • A. Lepskiy

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Thanks for you attention

alex.lepskiy@gmail.com http://lepskiy.ucoz.com

  • A. Lepskiy

(HSE) Coherence of the Conflict BELIEF 2018 21 / 21