On Internal Conflict as an External Conflict of a Decomposition of - - PowerPoint PPT Presentation

On Internal Conflict as an External Conflict

• f 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 conflict: entropy approach [H¨

• hle 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 2X be a powerset of X. The mass function is a set function m : 2X → [0, 1] that satisfies the conditions m(∅) = 0,

• A⊆X m(A) = 1.

The subset A ∈ 2X 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 FA = (A, 1), A ∈ 2X and ηA be a categorical BF corresponding to BE FA = (A, 1), ηX 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 F1 = (A1, m1) and F2 = (A2, m2). We have a question about a conflict between these BE. Historically, the conflict measure K0(F1, F2) associated with Dempster’s rule is the first among conflict measures: K0 = K0(F1, F2) =

• B∩C=∅,

B∈A1,C∈A2

m1(B)m2(C). If K0 = 1, then we have the following Dempster’s rule for combining

• f two BE:

mD(A) = 1 1 − K0

• B∩C=A m1(B)m2(C),

A = ∅, mD(∅) = 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: mDP(A) =

• B∪C=A m1(B)m2(C),

A ∈ 2X. (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 Fi = (Ai, mi), i = 1, ..., l with the help of some combining rule R: F = R(F1, ..., Fl). Therefore we can estimate the internal conflict by the formula KR

in(F) = K(F1, ..., Fl), assuming that F = R(F1, ..., Fl),

where K is some fixed (external) conflict measure, R is a fixed combining rule. Since the equation F = R(F1, ..., Fl) has many solutions then we can consider the optimization problem of finding the largest K

R in(F) and smallest KR in(F) conflicts:

K

R in(F) = arg max F =R(F1,...,Fl)

K(F1, ..., Fl), KR

in(F) = arg min F =R(F1,...,Fl)

K(F1, ..., Fl).

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

Let Sn ={(si)n

i=1 : si ≥ 0, n i=1 si =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 Fi = (Ai, mi) ∈ F(X), i = 1, 2, that satisfy the condition K0(F1, F2) =

• B∩C=∅,

B∈A1,C∈A2

m1(B)m2(C) → max (min) (2) with constraints (m1(B))B∈A1 ∈ S|A1|, (m2(C))C∈A2 ∈ S|A2|, (3) (1 − K0(F1, F2)) m(A) =

• B∩C=A,

B∈A1,C∈A2

m1(B)m2(C), A ∈ A. (4)

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 KD

in(F) = 0 and this

value is achieved on the pair F1 = F, F2 = FX. In the same time we have K

D in(F) = 1 and this value achieved for such

Fi = (Ai, mi) ∈ F(X), i = 1, 2, that B ∩ C = ∅ ∀B ∈ A1, ∀C ∈ A2. Therefore, in general formulation the problem of finding K

D in(F) and

KD

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(Fi), i = 1, 2, (5) where f : F(X) → [0, 1] is a some imprecision index. For example, it may be the generalized Hartley measure: f(F) = 1 ln |X|

• A∈A m(A) ln |A|.

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 F1 = F and F2 = FX since f(FX) = 1. Therefore we have always KD

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 g = m0ηX + n

i=1 miη{xi},

(mi)n

i=0 ∈ Sn+1.

Let us assume that Dempster’s rule is used to combine of BFs. In this case combinable BFs g1 and g2 should have the form g1 = α0ηX + n

i=1 αiη{xi},

g2 = β0ηX + n

i=1 βiη{xi}.

Then K0(g1, g2) = (1 − α0)(1 − β0) − n

i=1 αiβi.

(6)

Alexander Lepskiy (HSE) On Internal Conflict BELIEF 2016 11 / 22

Decomposition with Dempster’s Rule

The conditions (3)-(4) have the following form (αi)n

i=0 ∈ Sn+1,

(βi)n

i=0 ∈ Sn+1,

(7)   1−(1−α0)(1−β0)+

n

• j=1

αjβj   mi =αiβi + αiβ0+α0βi, i=1, ..., n, (8) The condition (5) for linear imprecision index has the form m0 ≤ α0, m0 ≤ β0. (9) Thus, the problem of finding the largest internal conflict K

D 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

Decomposition with Disjunctive rule

Decomposition of Evidence with the Help of Disjunctive Consensus Rule

The following estimation holds for disjunctive consensus rule and any linear imprecision index f: f(F) ≥ f(Fi), i = 1, 2, (10) i.e. imprecision of evidence is not reduced after the application of this combining rule. These inequalities reflect the pessimism of disjunctive consensus rule. If the one evidence states that true alternative belongs to the set A and another evidence states that the true alternative belongs to the set B then true alternative should be belong to the set A ∪ B after combining of these evidence with the help of disjunctive consensus rule. Thus, we have a problem of finding of bodies of evidence having the largest (smallest) conflict (2) and satisfying constraints (1), (3), (10).

Alexander Lepskiy (HSE) On Internal Conflict BELIEF 2016 13 / 22

Internal conflict for |X| = 2

Internal Conflict for |X| = 2. Decomposition with the Help of Dempster’s Rule

Let X = {x1, x2}. Then information is described by a BF g = m0ηX + m1η{x1} + m2η{x2} with (mi)2

i=0 ∈ S3. Since KD in(F) = 0,

then we will find the maximum of the function K0(g1, g2) = α1β2 + α2β1 with constrains (1 − αi)(1 − βi) = (1 − α1β2 − α2β1)(1 − mi), i = 1, 2, (11) α1+α2 ≤ m1+m2, β1+β2 ≤ m1+m2, αi ≥ 0, βi ≥ 0, i = 1, 2. (12)

Alexander Lepskiy (HSE) On Internal Conflict BELIEF 2016 14 / 22

Internal conflict for |X| = 2

The solution of this problem is achieved on the border of Ω = {(α1, α2) ∈ [0, 1]2 : α1 + α2 ≤ m1 + m2} and K

D in = m1m2 (1−m1)(1−m2) = m1m2 (m0+m1)(m0+m2).

Level lines of K

D in are shown on Fig.

Alexander Lepskiy (HSE) On Internal Conflict BELIEF 2016 15 / 22

Internal conflict for |X| = 2

We have K

D in ≈ 1 if m0 ≪ min{m1, m2} (see Fig). In particular, the

last condition is fulfilled when m0 ≈ 0 and min{m1, m2} ≫ 0, i.e. the BF is close to probability measure but not a Dirac measure. Since KD

in(F) = 0, then the uncertainty of internal conflict will be maximum

in this case. At that the value K

D in is more when the distance

|m1 − m2| is less for one and the same value of m0. Conversely, we have K

D in ≈ 0 (and hence KD in ≈ 0) if the BF is either

close to the Dirac measure m1 ≈ 1 ∨ m2 ≈ 1, or it is closer to the vacuous BF ηX (m0 ≈ 1).

Alexander Lepskiy (HSE) On Internal Conflict BELIEF 2016 16 / 22

Internal conflict for |X| = 2

Internal Conflict for |X| = 2. Decomposition with the Help of Disjunctive Consensus Rule

We should find the minimum (maximum) of K0(g1, g2) = α1β2 + α2β1 with constraints m1 = α1β1, m2 = α2β2, α1 + α2 ≥ m1 + m2, β1 + β2 ≥ m1 + m2, (αi)2

i=0 ∈ S3,

(βi)2

i=0 ∈ S3.

Problem is reduced to finding minimum (maximum) of the function K0 = α1 α2 m2 + α2 α1 m1 in the set Ω1(m1, m2) =

• (α1, α2) ∈ (0, 1]2 : α1 + α2 ≤ 1, m1

α1 + m2 α2 ≤ 1

• .

Alexander Lepskiy (HSE) On Internal Conflict BELIEF 2016 17 / 22

Internal conflict for |X| = 2

We have KDP

in (F) = (K0)min = 2√m1m2,

K

DP in

= (K0)max = m0 = 1−m1−m2. Let M =

• (m1, m2) ∈
• S2 : Ω1(m1, m2) = ∅
• .

The level lines are shown in Fig. for K = KDP

in

and K = K

DP in

• n M.

Alexander Lepskiy (HSE) On Internal Conflict BELIEF 2016 18 / 22

Internal conflict for |X| = 2

We have KDP

in (F) ≈ 0 and K DP in (F) ≈ 1 if F ≈ FX. In this case the

uncertainty of estimating conflict is maximal. If m0 ≈ 0, then the BF is close to a Dirac measure and K

DP in

≈ 0 in this case (and consequently, KDP

in

≈ 0). The estimation of internal conflict results in a unique KDP

in

= K

DP in

= 2√m1

• 1 − √m1
• , 0 < m1 < 1, on the curve

√m1 + √m2 = 1. In particular, this unique value is maximal and it is equal to 0.5 for BF g = 1

2ηX + 1 4η{x1} + 1 4η{x2}.

It is easy to show also, that K

D in(m1, m2) < KDP in (m1, m2) for all

(m1, m2) ∈ Ω1. This means that the estimation of an internal conflict

• btained with the help of optimistic Dempster’s rule is always less than

the estimation of an internal conflict obtained with the help of pessimistic disjunctive consensus rule.

Alexander Lepskiy (HSE) On Internal Conflict BELIEF 2016 19 / 22

Summary and Conclusion

Summary and Conclusion

It is shown that: interval estimations of internal conflict obtained with the help of decomposition by Dempster’s rule and disjunctive consensus rule do not intersect; in the case of decomposition by Dempster’s rule, the greatest uncertainty (0 ≤ KD

in ≤ 1) is achieved for the BF close to a

probability measure but not close to a Dirac measure; the value KD

in ≈ 0 is achieved for BF close to a Dirac measure either it is

close to the vacuous BF; in the case of decomposition by disjunctive consensus rule, the greatest uncertainty (0 ≤ KDP

in

≤ 1) is achieved for a vacuous BF F = ηX; the value KDP

in

≈ 0 is achieved for a Dirac measure.

Alexander Lepskiy (HSE) On Internal Conflict BELIEF 2016 20 / 22

References

Bronevich A., Lepskiy A.: Imprecision indices: axiomatic, properties and

• applications. Intern. J. of General Systems. 44(7-8), 812–832 (2015)

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) Daniel, M.: Conflict between Belief Functions: a New Measure Based on their Non-Conflicting Parts. In: Cuzzolin, F. (ed.) BELIEF 2014. LNCS, vol. 8764,

• pp. 321–330. Springer (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., Maupin, P.: Distances in evidence theory: Comprehensive survey and generalizations. Intern. J. of Approx. Reas. 53, 118–145 (2012) Lepskiy, A.: General Schemes of Combining Rules and the Quality Characteristics of Combining. Cuzzolin, F. (ed.) BELIEF 2014, LNAI, vol. 8764, pp. 29–38. Springer (2014) Martin, A.: About Conflict in the Theory of Belief Functions. In: Denoeux, T., Masson M-H. (eds.) BELIEF, AISC, vol.164, pp.161–168. Springer (2012) Schubert, J.: The Internal Conflict of a Belief Function. In: Denoeux, T., Masson M-H. (eds.) BELIEF 2012, AISC, vol. 164, pp. 169–177. Springer (2012)

Alexander Lepskiy (HSE) On Internal Conflict BELIEF 2016 21 / 22

Thanks for you attention

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

Alexander Lepskiy (HSE) On Internal Conflict BELIEF 2016 22 / 22