On Internal Conflict as an External Conflict of a Decomposition of Evidence Alexander Lepskiy National Research University - Higher School of Economics, Moscow, Russia The 4th International Conference on Belief Functions - BELIEF 2016, September 21-23, 2016, Prague, Czech Republic Alexander Lepskiy (HSE) On Internal Conflict BELIEF 2016 1 / 22
Preamble External and Internal Conflicts Conflictness is an important a priori characteristic of combining rules in the belief functions theory. The conflict of pieces of evidence characterizes the information inconsistency given by corresponding bodies of evidence. Conflictness of single evidence is considered together with the conflict between the bodies of evidence. In the first case we talk about the external conflict , in the second case we talk about the internal conflict . For example, we have the following evidence in which a large internal conflict is observed: the value of the company shares will be tomorrow in the interval [0,10] or [30,35]. Alexander Lepskiy (HSE) On Internal Conflict BELIEF 2016 2 / 22
Preamble Estimation of External and Internal Conflicts There are several approaches to the estimation of external conflict : axiomatic approach [Martin 2012, Destercke & Burger 2013, Bronevich et al. 2015]; metric approach [Jousselme et al. 2001, Jousselme & Maupin 2012, Liu 2006]; structural approach [Martin 2012]; algebraic approach [Lepskiy 2013] etc. There are the following approaches to the estimation of internal conflic t: entropy approach [H¨ ohle 1982, Yager 1983, Klir& Ramer 1990 etc]; axiomatic approach [Bronevich & Klir 2010]; algebraic approach [Daniel 2010]; decompositional approach [Schubert 2012] etc. Alexander Lepskiy (HSE) On Internal Conflict BELIEF 2016 3 / 22
Preamble The Main Idea of Decompositional Approach We will consider another approach, which also uses the idea of decomposition for definition of internal conflict. The following assumption is the basis of this approach. Evidence with a great internal conflict has been obtained as a result of aggregation information from several different sources with the help of some combining rule. Then the (external) conflict of the decomposed set of evidence can be regarded as an internal conflict of the original evidence. The decomposition result is ambiguous. Therefore we can talk only about the upper and lower estimates of the internal conflict in this case. In addition, it is necessary to introduce some additional restrictions in order to the result is not trivial or degenerate. Alexander Lepskiy (HSE) On Internal Conflict BELIEF 2016 4 / 22
Belief function and combining rules Belief Functions Theory Let X be a finite set and 2 X be a powerset of X . The mass function is a set function m : 2 X → [0 , 1] that satisfies the conditions m ( ∅ ) = 0, � A ⊆ X m ( A ) = 1. The subset A ∈ 2 X is called a focal element , if m ( A ) > 0. Let A = { A } be a set of all focal elements of evidence. The pair F = ( A , m ) is called a body of evidence (BE) . Let F ( X ) be a set of all BE on X . If F = ( A , m ) be a BE then � g ( B ) = A ⊆ B,A ∈A m ( A ) be a belief function (BF) corresponding to BE F . Let F A = ( A, 1), A ∈ 2 X and η A be a categorical BF corresponding to BE F A = ( A, 1), η X is called a vacuous BF . Alexander Lepskiy (HSE) On Internal Conflict BELIEF 2016 5 / 22
Belief function and combining rules Conflict Measure Let us have two BE F 1 = ( A 1 , m 1 ) and F 2 = ( A 2 , m 2 ). We have a question about a conflict between these BE. Historically, the conflict measure K 0 ( F 1 , F 2 ) associated with Dempster’s rule is the first among conflict measures: � K 0 = K 0 ( F 1 , F 2 ) = m 1 ( B ) m 2 ( C ) . B ∩ C = ∅ , B ∈A 1 ,C ∈A 2 If K 0 � = 1, then we have the following Dempster’s rule for combining of two BE: 1 � A � = ∅ , m D ( ∅ ) = 0 . m D ( A ) = B ∩ C = A m 1 ( B ) m 2 ( C ) , 1 − K 0 Dubois and Prade’s disjunctive consensus rule is a dual rule to Dempster’s rule in some sense. This rule is defined by a formula: � A ∈ 2 X . m DP ( A ) = B ∪ C = A m 1 ( B ) m 2 ( C ) , (1) Alexander Lepskiy (HSE) On Internal Conflict BELIEF 2016 6 / 22
Decomposition of Evidence The General Idea of Decomposition In general case we can assume that some evidence describing with the help of BE F = ( A , m ) has a great internal conflict, if its information source is a heterogeneous. In this case we can consider that the BE F = ( A , m ) is a result of combining of several BE F i = ( A i , m i ), i = 1 , ..., l with the help of some combining rule R : F = R ( F 1 , ..., F l ). Therefore we can estimate the internal conflict by the formula K R in ( F ) = K ( F 1 , ..., F l ) , assuming that F = R ( F 1 , ..., F l ) , where K is some fixed (external) conflict measure, R is a fixed combining rule. Since the equation F = R ( F 1 , ..., F l ) has many solutions then we can consider the optimization problem of finding the R in ( F ) and smallest K R largest K in ( F ) conflicts: R K R K in ( F ) = arg max K ( F 1 , ..., F l ) , in ( F ) = arg min K ( F 1 , ..., F l ) . F = R ( F 1 ,...,F l ) F = R ( F 1 ,...,F l ) Alexander Lepskiy (HSE) On Internal Conflict BELIEF 2016 7 / 22
Decomposition with Dempster’s Rule Decomposition of Evidence with the Help of Dempster’s Rule i =1 : s i ≥ 0 , � n Let S n = { ( s i ) n i =1 s i =1 } be a n -dimensional simplex. Then optimization problems for Dempster’s rule and l = 2 have the following formulation. We have to find the BE F i = ( A i , m i ) ∈ F ( X ), i = 1 , 2, that satisfy the condition � m 1 ( B ) m 2 ( C ) → max K 0 ( F 1 , F 2 ) = (min) (2) B ∩ C = ∅ , B ∈A 1 ,C ∈A 2 with constraints ( m 1 ( B )) B ∈A 1 ∈ S |A 1 | , ( m 2 ( C )) C ∈A 2 ∈ S |A 2 | , (3) � (1 − K 0 ( F 1 , F 2 )) m ( A ) = m 1 ( B ) m 2 ( C ) , A ∈ A . (4) B ∩ C = A, B ∈A 1 ,C ∈A 2 Alexander Lepskiy (HSE) On Internal Conflict BELIEF 2016 8 / 22
Decomposition with Dempster’s Rule Restrictions Related to the Optimistic Dempster’s Rule Note, that in the case of the general formulation K D in ( F ) = 0 and this value is achieved on the pair F 1 = F , F 2 = F X . D In the same time we have K in ( F ) = 1 and this value achieved for such F i = ( A i , m i ) ∈ F ( X ), i = 1 , 2, that B ∩ C = ∅ ∀ B ∈ A 1 , ∀ C ∈ A 2 . D Therefore, in general formulation the problem of finding K in ( F ) and K D in ( F ) is not meaningful. At the same time, Dempster’s rule is an optimistic rule in the following sense. If one evidence argues that the true alternative belongs to the set A , and the other – to the set B , then after combination of evidence in accordance with Dempster’s rule we get that the true alternative belong to the set A ∩ B . Alexander Lepskiy (HSE) On Internal Conflict BELIEF 2016 9 / 22
Decomposition with Dempster’s Rule Imprecision Index In general form the condition of optimism of Dempster’s rule may be described by inequalities f ( F ) ≤ f ( F i ) , i = 1 , 2 , (5) where f : F ( X ) → [0 , 1] is a some imprecision index. For example, it may be the generalized Hartley measure : 1 � f ( F ) = A ∈A m ( A ) ln | A | . ln | X | It is known that the estimation (5) is always true for any linear imprecision index f and non-conflicting set of evidence. Note that the conditions (5) are performed for the BE F 1 = F and F 2 = F X since f ( F X ) = 1. Therefore we have always K D in ( F ) = 0. Then the problem can be put to find BE with the largest conflict (2) and satisfying the conditions (3)-(5). Alexander Lepskiy (HSE) On Internal Conflict BELIEF 2016 10 / 22
Decomposition with Dempster’s Rule Example It is necessary to estimate the internal conflict of evidence given by a BF � n ( m i ) n g = m 0 η X + i =0 ∈ S n +1 . i =1 m i η { x i } , Let us assume that Dempster’s rule is used to combine of BFs. In this case combinable BFs g 1 and g 2 should have the form � n � n g 1 = α 0 η X + g 2 = β 0 η X + i =1 α i η { x i } , i =1 β i η { x i } . Then � n K 0 ( g 1 , g 2 ) = (1 − α 0 )(1 − β 0 ) − (6) i =1 α i β i . Alexander Lepskiy (HSE) On Internal Conflict BELIEF 2016 11 / 22
Decomposition with Dempster’s Rule The conditions (3)-(4) have the following form ( α i ) n ( β i ) n i =0 ∈ S n +1 , i =0 ∈ S n +1 , (7) n � 1 − (1 − α 0 )(1 − β 0 )+ α j β j m i = α i β i + α i β 0 + α 0 β i , i =1 , ..., n, (8) j =1 The condition (5) for linear imprecision index has the form m 0 ≤ α 0 , m 0 ≤ β 0 . (9) D Thus, the problem of finding the largest internal conflict K in has a form: it is necessary to find the largest value of the function (6) with constrains (7)-(9). Alexander Lepskiy (HSE) On Internal Conflict BELIEF 2016 12 / 22
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