Estimation of Conflict and Decreasing of Ignorance in - PowerPoint PPT Presentation

Estimation of Conflict and Decreasing of Ignorance in Dempster-Shafer Theory Alexander Lepskiy National Research University - Higher School of Economics, Moscow, Russia The 1 st International Conference on Information Technology and Quantitative Management, May 16 - 18, 2013, Suzhou, China Alexander Lepskiy (HSE) Conflict and Ignorance in DST ITQM 2013 1 / 23

Outline Outline of presentation 1 Theory of evidence Belief function and body of evidence Combining rules in Dempster-Shafer theory 2 Changing of ignorance after application of combining rules Imprecise indices Index of decreasing of ignorance 3 Conflict measure 4 Studying the relation between measure of conflict and index of decreasing of ignorance Statistical analyses Theoretical analyses 5 Summary and conclusion Alexander Lepskiy (HSE) Conflict and Ignorance in DST ITQM 2013 2 / 23

Theory of evidence Theory of evidence. Belief function and body of evidence Let X be a finite universal set and 2 X be the power set of X . Consider a belief measure g : 2 X → [0 , 1]. A belief function g is defined in evidence theory by a set function m g ( A ), called basic probability assignment (bpa): m g : 2 X → [0 , 1] , � m g ( ∅ ) = 0 , m g ( A ) = 1 . A ⊆ X B : B ⊆ A m g ( B ). Let the set of all belief measures on 2 X Then g ( A ) = � be denoted by Bel ( X ). Belief function g , and its dual, plausibility g ( A ) = 1 − g ( ¯ function ¯ A ), are considered together in evidence theory. Basic probability assignment m g may be computed by belief function g with help of so called Mobius transform of g : ( − 1) | B \ A | g ( A ) . � m g ( B ) = A : A ⊆ B Alexander Lepskiy (HSE) Conflict and Ignorance in DST ITQM 2013 3 / 23

Theory of evidence Combining rules in Dempster-Shafer theory The subset A ∈ 2 X is called by a focal element if m ( A ) > 0. Let A is a set of all focal elements. Then pair F = ( A , m ) is called a body of evidence . Let A ( g ) is the set of all focal elements and F ( g ) is the body of evidence related with belief function g . Suppose that we have two bodies of evidence ( A (1) , m (1) ) and ( A (2) , m (2) ) which are defined on the set X . In general a combining rule is a some operator R : Bel ( X ) × Bel ( X ) → Bel ( X ). Dempster’s rule (1967) 1 � m (1) ( A 1 ) m (2) ( A 2 ) , A � = ∅ , m D ( A ) = 1 − K A 1 ∩ A 2 = A � m (1) ( A 1 ) m (2) ( A 2 ) . m D ( ∅ ) = 0 , K = A 1 ∩ A 2 = ∅ The value K characterizes the amount of conflict of two sources of information. Alexander Lepskiy (HSE) Conflict and Ignorance in DST ITQM 2013 4 / 23

Theory of evidence Combining rules in Dempster-Shafer theory Discount rule (Shafer, 1976) m α ( A ) = (1 − α ) m ( A ) , A � = X ; m α ( X ) = α + (1 − α ) m ( X ) . The Dempster’s rule applies after discounting. The coefficient α ∈ [0 , 1] characterizes the degree of reliability of information: if α = 0 then source of information is absolutely reliable. If α = 1 then source of information is absolutely no reliable. Yager’s modified Dempster’s rule (1987) � m (1) ( A 1 ) m (2) ( A 2 ) , A ∈ 2 X , q ( A ) = A 1 ∩ A 2 = A m Y ( A ) = q ( A ) , A � = ∅ , X, m Y ( ∅ ) = q ( ∅ ) = K, m Y ( X ) = m Y ( ∅ ) + q ( X ) . The value q ( X ) = m (1) ( X ) m (2) ( X ) characterizes the amount of ignorance in two bodies of evidence ( A (1) , m (1) ) and ( A (2) , m (2) ). Alexander Lepskiy (HSE) Conflict and Ignorance in DST ITQM 2013 5 / 23

Theory of evidence Combining rules in Dempster-Shafer theory Inagaki’s unified combination rule (1991) m I ( A ) = q ( A )(1 + kq ( ∅ )) , A � = X, m I ( X ) = q ( X )(1 + kq ( ∅ )) + q ( ∅ )(1 + kq ( ∅ ) − k ) , where 0 ≤ k ≤ 1/(1 − q ( ∅ ) − q ( X )). If k = 0 then we have Yager’s rule. If k = 1/(1 − q ( ∅ )) then we get Dempster’s rule. Zhang’s center combination rule (1994) � r ( A 1 , A 2 ) m (1) ( A 1 ) m (2) ( A 2 ) , A ∈ 2 X , m Z ( A ) = A 1 ∩ A 2 = A where r ( A 1 , A 2 ) be a measure of intersection of sets A 1 and A 2 . For example r ( A 1 , A 2 ) = c | A 1 ∩ A 2 | | A 1 ∪ A | Jaccard similarity coefficient. Dubois and Prade’s disjunctive consensus rule (1992) � m (1) ( A 1 ) m (2) ( A 2 ) , A ∈ 2 X . m DP ( A ) = A 1 ∪ A 2 = A Alexander Lepskiy (HSE) Conflict and Ignorance in DST ITQM 2013 6 / 23

Quantity of information ignorance Quantity of information ignorance. Measure of uncertainty Let we know only that the “true” alternative is in a nonempty set B ⊆ X . This situation can be described by the non-additive measure (the so-called primitive belief function ) � 1 , B ⊆ A, η � B � ( A ) = B �⊆ A, 0 , A ⊆ X , B � = ∅ . Hartley’s measure H ( η � B � ) = log 2 | B | characterizes the degree of imprecision of the information about belonging of “true” alternative. Let g = � m g ( B ) η � B � be a belief function. Then B ∈ 2 X \{∅} generalized Hartley’s measure is defined by � m ( B )log 2 | B | . GH ( g ) = B ∈ 2 X \{∅} Alexander Lepskiy (HSE) Conflict and Ignorance in DST ITQM 2013 7 / 23

Quantity of information ignorance Imprecise indices Definition 1. A functional f : Bel ( X ) → [0 , 1] is called imprecision index if the following conditions are fulfilled: 1 if g be a probability measure then f ( g ) = 0; 2 f ( g 1 ) ≥ f ( g 2 ) for all g 1 , g 2 ∈ Bel ( X ) such that g 1 ≤ g 2 ; 3 f � � η � X � = 1. An imprecision index f on Bel ( X ) is called linear if for any linear combination � k j =1 α j g j ∈ Bel ( X ), , g j ∈ Bel ( X ), j = 1 , ..., k , we have �� k � = � k j =1 α j f ( g j ). f j =1 α j g j Alexander Lepskiy (HSE) Conflict and Ignorance in DST ITQM 2013 8 / 23

Quantity of information ignorance Proposition 1. The functional f : Bel ( X ) → [0 , 1] is a linear imprecision index on Bel ( X ) iff f ( g ) = � m g ( B ) µ f ( B ), where set function µ f satisfies B ∈ 2 X \{∅} the conditions: 1 µ f ( { x } ) = 0 for any x ∈ X ; 2 µ f ( X ) = f � � η � X � = 1; 3 µ f be a monotonic set function i.e. µ f ( B ′ ) ≤ µ f ( B ′′ ) if B ′ ⊆ B ′′ . Suppose that we have two bodies of evidence F ( g 1 ) = ( A (1) , m (1) ) and F ( g 2 ) = ( A (2) , m (2) ). These bodies of evidence corresponds belief functions g 1 and g 2 correspondingly. Let f : Bel ( X ) → [0 , 1] be a some linear imprecision index that estimates the degree of ignorance contained in the measure g . Suppose that we used some combining rule R for combining of evidence F ( g 1 ) and F ( g 2 ). As a result we get new belief function g = R ( g 1 , g 2 ). Then we have a question about amount of decreasing of ignorance after the using of combining rule R . Alexander Lepskiy (HSE) Conflict and Ignorance in DST ITQM 2013 9 / 23

Quantity of information ignorance Index of decreasing of ignorance The degree of such decreasing may be estimated with help of comparison f ( g ) with f ( g 1 ) and f ( g 2 ). For example we may introduce the following indices of decreasing of ignorance I R ( g i | g j ) = f ( g i ) − f ( R ( g i , g j )) , i, j ∈ { 1 , 2 } , I R ( g 1 , g 2 ) = min { I R ( g 1 | g 2 ) , I R ( g 2 | g 1 ) } . The decreasing of ignorance corresponded to the case of positivity of index I R ( g 1 , g 2 ). Alexander Lepskiy (HSE) Conflict and Ignorance in DST ITQM 2013 10 / 23

Quantity of information ignorance Some partial cases of evidence. Consensual evidences Let A (1) and A (2) are the two sets of focal elements satisfying the conditions: 1 A ′ ∩ A ′′ = ∅ , B ′ ∩ B ′′ = ∅ for all A ′ , A ′′ ∈ A (1) , B ′ , B ′′ ∈ A (2) ; 2 for every A ∈ A (1) exists a unique B ∈ A (2) such that A ∩ B � = ∅ ; 3 for every B ∈ A (2) exists a unique A ∈ A (1) such that A ∩ B � = ∅ . We will call this situation by a “consensual evidences” . Thus there is a one-to-one correspondence ϕ between the elements of sets A (1) and A (2) . ... A A B B 1 1 k k Alexander Lepskiy (HSE) Conflict and Ignorance in DST ITQM 2013 11 / 23

Quantity of information ignorance Some partial cases of evidence. Clarifying evidences If two bodies of evidence satisfy the conditions 1)-3) and the additional condition 4 A ⊆ ϕ ( A ) for all A ∈ A (1) then we will call this situation by “clarifying evidences” . A A ... k 1 B B k 1 Alexander Lepskiy (HSE) Conflict and Ignorance in DST ITQM 2013 12 / 23

Quantity of information ignorance Decreasing of ignorance. Dempster’s rule Proposition 2. Let F ( g 1 ) = ( A (1) , m (1) ) and F ( g 2 ) = ( A (2) , m (2) ) are the two bodies of evidence satisfying the conditions 1)-3). Then I D ( g 1 , g 2 ) > 0 if � � m (1) ( A ) ) ,m (2) ( ϕ ( A ) ) � m (1) ( A ) m (2) ( ϕ ( A ) ) > max A ∈A (1) µ f ( A ∩ ϕ ( A ))max . µ f ( ϕ ( A ) µ f ( A ) A ∈A (1) Corollary 1. Let two bodies of evidence F ( g 1 ) = ( A (1) , m (1) ) and F ( g 2 ) = ( A (2) , m (2) ) satisfy the conditions 1)-4). Then I D ( g 1 , g 2 ) > 0 if the following condition is true: � µ f ( A ) � � m (1) ( A ) m (2) ( ϕ ( A )) > max m (1) ( A ) µ f ( ϕ ( A )) , m (2) ( ϕ ( A )) A ∈A (1) max . A ∈A (1) Alexander Lepskiy (HSE) Conflict and Ignorance in DST ITQM 2013 13 / 23