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Estimation of Conflict and Decreasing

• f Ignorance in Dempster-Shafer Theory

Alexander Lepskiy

National Research University - Higher School of Economics, Moscow, Russia

The 1st 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 2X be the power set of X. Consider a belief measure g : 2X → [0, 1]. A belief function g is defined in evidence theory by a set function mg(A), called basic probability assignment (bpa): mg : 2X → [0, 1], mg(∅) = 0,

• A⊆X

mg(A) = 1. Then g(A) =

B: B⊆A mg(B). Let the set of all belief measures on 2X

be denoted by Bel(X). Belief function g, and its dual, plausibility function ¯ g(A) = 1 − g( ¯ A), are considered together in evidence theory. Basic probability assignment mg may be computed by belief function g with help of so called Mobius transform of g: mg(B) =

• A:A⊆B

(−1)|B\A|g(A).

Alexander Lepskiy (HSE) Conflict and Ignorance in DST ITQM 2013 3 / 23

Theory of evidence

Combining rules in Dempster-Shafer theory

The subset A ∈ 2X 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) mD(A) = 1 1 − K

• A1∩A2=A

m(1)(A1)m(2)(A2), A = ∅, mD(∅) = 0, K =

• A1∩A2=∅

m(1)(A1)m(2)(A2). 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) mα(A) = (1 − α)m(A), A = X; mα(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) q(A) =

• A1∩A2=A

m(1)(A1)m(2)(A2), A ∈ 2X, mY (A) = q(A), A = ∅, X, mY (∅) = q(∅) = K, mY (X) = mY (∅) + 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) mI(A) = q(A)(1 + kq(∅)), A = X, mI(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) mZ(A) =

• A1∩A2=A

r(A1, A2)m(1)(A1)m(2)(A2), A ∈ 2X, where r(A1, A2) be a measure of intersection of sets A1 and A2. For example r(A1, A2) = c|A1∩A2|

|A1∪A| Jaccard similarity coefficient.

Dubois and Prade’s disjunctive consensus rule (1992) mDP (A) =

• A1∪A2=A

m(1)(A1)m(2)(A2), A ∈ 2X.

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) ηB(A) =

• 1,

B ⊆ A, 0, B ⊆ A, A ⊆ X, B = ∅. Hartley’s measure H(ηB) = log2 |B| characterizes the degree of imprecision of the information about belonging of “true”

• alternative. Let g =
• B∈2X\{∅}

mg(B)ηB be a belief function. Then generalized Hartley’s measure is defined by GH (g) =

• B∈2X\{∅}

m(B)log2 |B| .

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(g1) ≥ f(g2) for all g1, g2 ∈ Bel(X) such that g1 ≤ g2; 3 f

• ηX
• = 1.

An imprecision index f on Bel(X) is called linear if for any linear combination k

j=1 αjgj ∈ Bel(X), , gj ∈ Bel(X), j = 1, ..., k, we have

f k

j=1 αjgj

• = k

j=1 αjf (gj).

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) =

• B∈2X\{∅}

mg(B)µf(B), where set function µf satisfies the conditions:

1 µf ({x}) = 0 for any x ∈ X; 2 µf(X) = f

• ηX
• = 1;

3 µf be a monotonic set function i.e. µf(B′) ≤ µf(B′′) if B′ ⊆ B′′.

Suppose that we have two bodies of evidence F(g1) = (A(1), m(1)) and F(g2) = (A(2), m(2)). These bodies of evidence corresponds belief functions g1 and g2 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(g1) and F(g2). As a result we get new belief function g = R(g1, g2). Then we have a question about amount

• f 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(g1) and f(g2). For example we may introduce the following indices of decreasing of ignorance IR(gi|gj) = f(gi) − f(R(gi, gj)), i, j ∈ {1, 2}, IR(g1, g2) = min {IR(g1|g2), IR(g2|g1)} . The decreasing of ignorance corresponded to the case of positivity of index IR(g1, g2).

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).

...

1

A

1

B

k

A

k

B

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”.

...

1

A

1

B

k

A

k

B

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(g1) = (A(1), m(1)) and F(g2) = (A(2), m(2)) are the two bodies of evidence satisfying the conditions 1)-3). Then ID(g1, g2) > 0 if

• A∈A(1)

m(1)(A)m(2)(ϕ(A) )> max

A∈A(1)µf(A∩ϕ(A))max

• m(1)(A)

µf(ϕ(A) ),m(2)(ϕ(A) ) µf(A)

• .

Corollary 1. Let two bodies of evidence F(g1) = (A(1), m(1)) and F(g2) = (A(2), m(2)) satisfy the conditions 1)-4). Then ID(g1, g2) > 0 if the following condition is true:

• A∈A(1)

m(1)(A)m(2)(ϕ(A))> max

A∈A(1)max

• m(1)(A)

µf(A) µf(ϕ(A)), m(2)(ϕ(A))

• .

Alexander Lepskiy (HSE) Conflict and Ignorance in DST ITQM 2013 13 / 23

Quantity of information ignorance

Decreasing of ignorance. Yager’s rule

Proposition 3. Let F(g1) = (A(1), m(1)) and F(g2) = (A(2), m(2)) are the two bodies of evidence satisfying the conditions 1)-3). Then IY (g1, g2) > 0 iff

• A∈A(1)

m(1)(A)m(2)(ϕ(A) ) (1−µf(A ∩ ϕ(A) )) > > max   

• A∈A(1)

m(1)(A) (1−µf(A) ),

• A∈A(1)

m(2)(ϕ(A) ) (1−µf(ϕ(A) ))    . Corollary 2. Let two bodies of evidence F(g1)=( A(1), m(1)) and F(g2)=( A(2), m(2)) satisfy the conditions 1)-4). Then the index of decreasing of ignorance IY (g1|g2)=f(g1)−f(Y (g1, g2)) will nonpositive for Yager’s rule.

Alexander Lepskiy (HSE) Conflict and Ignorance in DST ITQM 2013 14 / 23

Conflict measure

Conflict measure

Let F1 =(A(g1), m(1)), and F2 =(A(g2), m(2)), are the two bodies of evidence on X related with belief functions g1 and g2 correspondingly; r : 2X × 2X → [0, 1] be a some measure of intersection of sets that satisfy following conditions: 1) r(A, B)=r(B, A); 2) r(A, B)=0 if A ∩ B =∅; 3) r(A, A)=1, A = ∅. Definition 2. A functional cr : Bel(X) × Bel(X) → [0, 1] is called by measure of conflict with respect to r if the following condition are fulfilled:

1 cr(g1, g2) = cr(g2, g1) for all g1, g2 ∈ Bel(X); 2 cr(g′, g) ≥ cr(g′′, g) if F(g′) = F ∪ (A′, m), F(g′′) = F ∪ (A′′, m)

and r(A′, B) ≤ r(A′′, B) for all B ∈ A(g);

3 cr(g1, g2) = 1 if A ∩ B = ∅ for all A ∈ A(g1), B ∈ A(g2).

A measure of conflict cr is called by bilinear if cr(αg1+βg2, g)= αcr(g1, g)+βcr(g2, g) for all α, β ∈[0, 1], α+β =1, g, g1, g2 ∈Bel(X).

Alexander Lepskiy (HSE) Conflict and Ignorance in DST ITQM 2013 15 / 23

Conflict measure

General form of conflict measure

Proposition 4. A functional cr be a bilinear measure of conflict on Bel(X) × Bel(X) iff cr(g1, g2) =

• A∈A(g1),B∈A(g2)

γ(A, B)m(1)(A)m(2)(B), where γ(A, B)∈[0, 1] satisfy following conditions: a) γ(A, B)=γ(B, A); b) γ(A′, B)≥γ(A′′, B) if r(A′, B)≤r(A′′, B); c) γ(A, B)=1 if A ∩ B =∅. For example, the measure of conflict γ(A, B) = cr

• ηA, ηB
• = ϕ(r(A, B)), A, B = ∅, satisfy the conditions

a)-c) of Proposition 4 if ϕ is a nonincreasing function for which ϕ(1) = 0, ϕ(0) = 1 and r(A, B) =

|A∩B| min{|A|,|B|}.

In this case cr(g1, g2)=K if r(A, B)= 1, A∩B =∅, 0, A∩B =∅ be a primitive measure of intersection.

Alexander Lepskiy (HSE) Conflict and Ignorance in DST ITQM 2013 16 / 23

Relation between conflict and ignorance

Relation between measure of conflict and index of decreasing of ignorance. Statistical analyses

Let focal elements and basic probability assignments are generated by

• uniformly. The bodies of evidence have a capacity that is equal to two.

0.01 0.12 0.23 0.34 0.45 0.56 0.67 0.78 0.89 1 K

Figure: the histogram of conflict measure in general case

Alexander Lepskiy (HSE) Conflict and Ignorance in DST ITQM 2013 17 / 23

Relation between conflict and ignorance

Statistical analyses in general case

Figure: distribution of points (K, ID) when combining the two evidence in general case Figure: the dependence of the estimate of the probability PD(K)=P{ ID >0|K} from conflict in general case

Alexander Lepskiy (HSE) Conflict and Ignorance in DST ITQM 2013 18 / 23

Relation between conflict and ignorance

Statistical analyses in case of “clarifying evidence”

Figure: the histogram and function of density of conflict measure in case of clarifying evidences Figure: distribution of points (K, ID) when combining the two clarifying evidences

Alexander Lepskiy (HSE) Conflict and Ignorance in DST ITQM 2013 19 / 23

Relation between conflict and ignorance

Statistical analyses in case of “clarifying evidence”

Figure: the dependence of the estimate of the probability PD(K)=P{ ID >0|K} from conflict in case of clarifying evidences

Alexander Lepskiy (HSE) Conflict and Ignorance in DST ITQM 2013 20 / 23

Relation between conflict and ignorance

Theoretical analyses

Let IR(K) = sup {IR(g1, g2) : c(g1, g2) = K}, IR(K) = inf {IR(g1, g2) : c(g1, g2) = K}. Proposition 5. Let we combine two clarifying evidence of cardinality 2, the conflict measure is equal c(g1, g2) = K and the index of decreasing of ignorance computed with help of generalized normalized Hartley’s measure. Then we have ID(K) ≤ K, ID(K) ≥ −K. Proposition 6. Let two clarifying bodies of evidence have a cardinality which equal two and their basic probability assignments have uniform distributions. Then the random variable – conflict measure K has a probability density equal to hK(x) = − ln |2x − 1|, x ∈ [0, 1/2) ∪ (1/2, 1].

Alexander Lepskiy (HSE) Conflict and Ignorance in DST ITQM 2013 21 / 23

Summary and conclusion

Summary and conclusion

It is shown that imprecise indices may be used for estimation of quantity of ignorance in body of evidence in framework of DST The index of decreasing of ignorance was introduced. The conditions were found which guarantee the decreasing of ignorance after the using of combining rule for bodies of evidence of special types The measure of conflict two evidence was introduced by axiomatically It is shown that quality of using of combining rules may be characterized by values of index of decreasing of ignorance and measure of conflict The relation between the index of decreasing of ignorance and measure of conflict was researched

Alexander Lepskiy (HSE) Conflict and Ignorance in DST ITQM 2013 22 / 23

Thanks for you attention

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

Alexander Lepskiy (HSE) Conflict and Ignorance in DST ITQM 2013 23 / 23