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
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
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 of CM; the structure of conjunctive and disjunctive CMs. A. Lepskiy (HSE) Coherence of the Conflict BELIEF 2018 3 / 21
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
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 = { A i } 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 F A = ( A, 1)) if it has only one focal element; BE F X = ( X, 1) is said to be vacuous ; if F j = ( A j , m j ) ∈ F ( X ) and � j α j = 1, α j ∈ [0 , 1] ∀ j , then F = � j α j F j = ( A , m ) ∈ F ( X ), where A = � j A j , m ( A ) = � j α j m j ( A ); we have F = � A ∈A m ( A ) F A ∀ F = ( A , m ). A. Lepskiy (HSE) Coherence of the Conflict BELIEF 2018 5 / 21
Evidence Theory and Combining Rules Conjunctive Combining Rules Let we have two BE F 1 = ( A 1 , m 1 ) and F 2 = ( A 2 , m 2 ) 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 D 0 ( F 1 , F 2 ) is considered � m D 0 ( A ) = A ∈ 2 X . B ∩ C = A m 1 ( B ) m 2 ( C ) , The conjunctive conflict measure K D ( F 1 , F 2 ) = m D 0 ( ∅ ) characterizes the amount of conflict between two sources of information described by the F 1 and F 2 . If K D � = 1, then the classical Dempster rule for combining of two BE can be defined: 1 m D ( A ) = 1 − K D m D 0 ( A ) , m D ( ∅ ) = 0 . A � = ∅ , A. Lepskiy (HSE) Coherence of the Conflict BELIEF 2018 6 / 21
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: m DP ( A ) = � A ∈ 2 X . B ∪ C = A m 1 ( B ) m 2 ( C ) , The duality relation is true by analogy with De Morgan’s law: � � D 0 ( F 1 , F 2 ) = DP F 1 , F 2 , where the complement BE F = � A ∈A m ( A ) F A , if F = � A ∈A m ( A ) F A . A. Lepskiy (HSE) Coherence of the Conflict BELIEF 2018 7 / 21
Axioms of Conflict Measures Axioms of Conflict Measures In general, it is desirable that the CM K ( F 1 , F 2 ) between BE satisfies the following conditions (axioms) [Martin 2012, Destercke & Burger 2013, Bronevich et al. 2015, ...] : A1 : 0 ≤ K ( F 1 , F 2 ) ≤ 1 ∀ F 1 , F 2 ∈ F ( X ) (non-negativity and normalization); A2 : K ( F 1 , F 2 ) = K ( F 2 , F 1 ) ∀ F 1 , F 2 ∈ 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 ( F X , F ) = 0 ∀ F ∈ F ( X ) (ignorance is bliss); A6 : K ( F A , F B ) = 1, if A ∩ B = ∅ . A. Lepskiy (HSE) Coherence of the Conflict BELIEF 2018 8 / 21
CMs Induced by Combining Rules CM Induced by Conjunctive Rule � K D ( F 1 , F 2 ) = B ∩ C = ∅ m 1 ( B ) m 2 ( 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 � K D ( F 1 , F 2 ) = 1 − B m 1 ( B ) m 2 ( B ) . In this case the measure K D is close to 2 �√ m 1 − √ m 2 � � 2 � � K H ( F 1 , F 2 ) = 1 − � 1 m 1 ( B ) m 2 ( B ) = . √ B � √ m 1 − √ m 2 1 � is Hellinger distance between probability � � The √ 2 distributions m 1 and m 2 . The CM K D is close also to � K M ( F 1 , F 2 ) = 1 − B min { m 1 ( B ) , m 2 ( B ) } [Bronevich & Rozenberg 2016] on the disjoint belief structure. A. Lepskiy (HSE) Coherence of the Conflict BELIEF 2018 9 / 21
CMs Induced by Combining Rules CM Induced by Disjunctive Consensus Rule K DP ( F 1 , F 2 ) = m DP ( X ) = � B ∪ C = X m 1 ( B ) m 2 ( C ) . Proposition 1 K DP ( F 1 , F 2 ) = K D ( F 1 , F 2 ); 2 if F 1 = ( A 1 , m 1 ), F 2 = ( A 2 , m 2 ) and A 1 = ¯ A 2 = { ¯ A : A ∈ A 2 } , then K D ( F 1 , F 2 ) = 1 ⇒ K DP ( F 1 , F 2 ) = 1. Note that the measure 1 − K DP 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
CMs Induced by Combining Rules The disjunctive CM satisfies the following conditions: A1 : 0 ≤ K DP ( F 1 , F 2 ) ≤ 1 ∀ F 1 , F 2 ∈ F ( X ) (non-negativity and normalization); A2 : K DP ( F 1 , F 2 ) = K DP ( F 2 , F 1 ) ∀ F 1 , F 2 ∈ F ( X ) (symmetry); A4 ′ : K DP ( F ′ , F ) ≤ K DP ( 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 ′ : K DP ( 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 ′ : K DP ( F A , F A ) = 1. A. Lepskiy (HSE) Coherence of the Conflict BELIEF 2018 11 / 21
CMs Induced by Combining Rules Example Let F 1 = α 1 F { x 1 } + α 2 F { x 2 } + (1 − α 1 − α 2 ) F { x 1 ,x 2 } , F 2 = β 1 F { x 1 } + β 2 F { x 2 } + (1 − β 1 − β 2 ) F { x 1 ,x 2 } be two BE on X = { x 1 , x 2 } . Then K DP ( F 1 , F 2 ) = 1 − ( α , β ) , K D ( F 1 , F 2 ) = | [ α , β − ] | , where α = ( α 1 , α 2 ), β = ( β 1 , β 2 ), β − = ( β 1 , − β 2 ), [ · , · ] is a vector product,( · , · ) is a scalar product. We have K D ( F 1 , F 2 ) ≤ K DP ( F 1 , F 2 ) . In particular, K D ( F 1 , F 2 ) = 1 ⇔ α = (1 , 0) , β = (0 , 1) and vice versa; K DP ( F 1 , F 2 ) = 1 ⇔ α ⊥ β ⇔ α = ( α, 0) ∧ β = (0 , β ) or vice versa. Analogically, K D ( F 1 , F 2 ) = 0 ⇔ α = ( α, 0) , β = ( β, 0) and vice versa; K DP ( F 1 , F 2 ) = 0 ⇔ α = (1 , 0) ∧ β = (1 , 0) or vice versa. A. Lepskiy (HSE) Coherence of the Conflict BELIEF 2018 12 / 21
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 ( F 1 , F 2 ) ≤ K ( R ( F 1 , F 2 ) , F i ) , i = 1 , 2 ∀ F 1 , F 2 ∈ 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 D 0 , which was considered in [Loudahi et al. 2014] : K ( D 0 ( F 1 , F 3 ) , D 0 ( F 2 , F 3 )) ≤ K ( F 1 , F 2 ). A. Lepskiy (HSE) Coherence of the Conflict BELIEF 2018 13 / 21
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