Conjunctive Rules in the Theory of Belief Functions and Their Justification through Decisions Models Andrey G. Bronevich 1 , Igor N. Rozenberg 2 1 National Research University ”Higher School of Economics”, Moscow, Russia 2 JSC Research, Development and Planning Institute for Railway Information Technology, Automation and Telecommunication, Moscow, Russia 4th International Conference on Belief Functions, 21-23 September 2016, Prague, Czech Republic (HSE, Moscow, Russia) Conjunctive Rules Belief-2016 1 / 14
Main Results Decisions models can be viewed as partial orders on real valued functions, and the conjunctive rule can be understood as the union of these orders. There is the contradiction among information sources if there is no order that contains this union. The disjunctive rule can be viewed as the intersection of orders and it is always exists. We consider the measure of contradiction connected with the conjunctive rule and give a number of axioms that define it uniquely. (HSE, Moscow, Russia) Conjunctive Rules Belief-2016 2 / 14
Notation and some facts from the theory of belief functions Let X be a finite set and 2 X be the powerset of its subsets. A set function Bel : 2 X → [0 , 1] is called a belief function if � Bel ( A ) = m ( B ) , B ∈ 2 X | B ⊆ A where m : 2 X → [0 , 1] called the basic belief assignment (bba) is such that B ∈ 2 X m ( B ) = 1. � A set B is called a focal element if m ( B ) > 0. The set of all focal elements is called the body of evidence . If the body of evidence contains only one focal element B , then the corresponding belief function η � B � is called categorical and � 1 , B ⊆ A, η � B � ( A ) = B �⊆ A. 0 , (HSE, Moscow, Russia) Conjunctive Rules Belief-2016 3 / 14
Notation and some facts from the theory of belief functions Any belief function Bel on 2 X can represented as a sum of categorical belief functions as � Bel = m ( B ) η � B � . B ∈ 2 X A belief function is called normalized if Bel ( ∅ ) = 0. The value Bel ( ∅ ) shows the amount of contradiction in information. Notation M bel is the set of all normalized belief functions on 2 X and the set of all belief functions including non-normalized ones is denoted by ¯ M bel ; M pr is the set of all probability measures on 2 X , i.e. normalized belief functions, for which m ( A ) = 0 if | A | � 2. (HSE, Moscow, Russia) Conjunctive Rules Belief-2016 4 / 14
Aggregation rules in the theory of belief functions Let we have two sources of information described by belief functions Bel i = A ∈ 2 X m i ( A ) η � A � , i = 1 , 2. If sources are assume to be reliable, � then we can apply the conjunctive rule � Bel = m ( A, B ) η � A ∩ B � , A,B ∈ 2 X where a joint belief assignment m : 2 X × 2 X → [0 , 1] satisfies the following conditions: � � A ∈ 2 X m ( A, B ) = m 2 ( B ) , (1) � B ∈ 2 X m ( A, B ) = m 1 ( A ) . We get the classical conjunctive rule, if we assume that sources of information are independent, i.e. m ( A, B ) = m 1 ( A ) m 2 ( B ), A, B ∈ 2 X . (HSE, Moscow, Russia) Conjunctive Rules Belief-2016 5 / 14
Aggregation rules in the theory of belief functions Dempster’s and Yager’s aggregation rules 1 1 Dempster’s rule: Bel = � m 1 ( A ) m 2 ( B ) η � A ∩ B � , where 1 − k A ∩ B � = ∅ � k = m 1 ( A ) m 2 ( B ); A ∩ B = ∅ 2 Yager’s rule: Bel = � m 1 ( A ) m 2 ( B ) η � A ∩ B � + kη � X � , where k is A ∩ B � = ∅ defined as in item 1; are closely related to the classical conjunctive rule. As one can see they show how the result of the classical conjunctive rule can be transformed to the normalized belief function. (HSE, Moscow, Russia) Conjunctive Rules Belief-2016 6 / 14
Aggregation rules in the theory of belief functions The disjunctive rule is defined by � Bel = m ( A, B ) η � A ∪ B � , A,B ∈ 2 X where the joint belief assignment obeying the same conditions (1) as for the conjunctive rule. It is used if at least one source of information is reliable. If the sources of information are independent, then the result of disjunctive rule is Bel ( A ) = Bel 1 ( A ) Bel 2 ( A ) for all A ∈ 2 X . Let we have m sources of information described by belief functions Bel i , i = 1 , ..., m , and reliability of i -th source , i = 1 , ..., m , is m � evaluated by r i � 0 and r i = 1. i =1 m � Then the mixture rule is defined as Bel = r i Bel i . i =1 (HSE, Moscow, Russia) Conjunctive Rules Belief-2016 7 / 14
Decision models based on imprecise probabilities Notation. K is the set of all real valued functions on X . Assume that any decision is identified with a f ∈ K on X and the information is described by P ∈ M pr . Then the preference order ≺ on K , based on suspected utility n � f ( x i ) P ( { x i } ) , E P ( f ) = i =1 is f 1 ≺ f 2 iff E P ( f 1 ) < E P ( f 2 ). If information is described by Bel ∈ M bel or the corresponding credal set P = { P ∈ M pr | P � Bel } , then possible decision rules are a) f 1 ≺ f 2 iff E P ( f 1 ) < E P ( f 2 ) for all P ∈ P ; b) f 1 ≺ f 2 iff E P ( f 1 ) < E P ( f 2 ), where E P ( f ) = inf P ∈ P E P ( f ); c) f 1 ≺ f 2 iff ¯ E P ( f 1 ) < ¯ E P ( f 2 ), where ¯ E P ( f ) = sup E P ( f ); P ∈ P d) f 1 ≺ f 2 iff E P ( f 1 ) < E P ( f 2 ) and ¯ E P ( f 1 ) < ¯ E P ( f 2 ). (HSE, Moscow, Russia) Conjunctive Rules Belief-2016 8 / 14
Aggregation Rules and Decision Models Let we have m sources of information, and for each source i ∈ { 1 , ..., m } we obtain the preference order ρ i on K . Then 1 the conjunctive rule should give us an order ρ obeying the consensus condition ρ i ⊆ ρ , i = 1 , ..., m . 2 the disjunctive rule should give us an order ρ obeying the condition ρ i ⊇ ρ , i = 1 , ..., m . If ρ obeying 1) does not exist then we say that sources of information are contradictory . m The disjunctive rule always exists and it can be defined as ρ = � ρ i . i =1 (HSE, Moscow, Russia) Conjunctive Rules Belief-2016 9 / 14
Aggregation Rules and Decision Models Lemma 1 Let Bel ∈ M bel be the result of the conjunctive rule to belief functions Bel 1 , Bel 2 ∈ M bel . Let us consider preference orders ρ, ρ 1 , ρ 2 that correspond to belief functions Bel, Bel 1 , Bel 2 by decision rule a). Then the preference order ρ for Bel agrees with orders ρ 1 and ρ 2 . Lemma 2 Let Bel ∈ M bel be the result of the disjunctive rule to belief functions Bel 1 , Bel 2 ∈ M bel . Then ρ i ⊇ ρ , i = 1 , 2 . Proposition 1 Sources of information described by belief functions Bel 1 , Bel 2 ∈ M bel are not contradictory iff P ( Bel 1 ) ∩ P ( Bel 2 ) � = ∅ . In this case there is a conjunctive rule with the result Bel ∈ M bel . (HSE, Moscow, Russia) Conjunctive Rules Belief-2016 10 / 14
Measure of contradiction Let R ( Bel 1 , Bel 2 ) be the set of possible belief functions obtained by the conjunctive rules applied to Bel 1 , Bel 2 ∈ M bel . Then the measure of contradiction Con : M bel × M bel → [0 , 1] is defined as Con ( Bel 1 , Bel 2 ) = inf { Bel ( ∅ ) | Bel ∈ R ( Bel 1 , Bel 2 ) } . Specialization order on M bel Let Bel 1 , Bel 2 ∈ ¯ M bel , then Bel 1 � Bel 2 iff there are representations N N N � � � Bel 1 = a i η � A i � and Bel 2 = a i η � B i � , such that a i = 1, a i � 0, i =1 i =1 i =1 B i ⊆ A i , i = 1 , ..., n . Remark Bel 1 � Bel 2 implies Bel 1 � Bel 2 ( Bel 1 ( A ) � Bel 2 ( A ) for all A ∈ 2 X ), but the opposite is not true in general. (HSE, Moscow, Russia) Conjunctive Rules Belief-2016 11 / 14
Properties of Con ( Bel 1 , Bel 2 ) A1. Con ( Bel 1 , Bel 2 ) = 0 for Bel 1 , Bel 2 ∈ M bel iff P ( Bel 1 ) ∩ P ( Bel 2 ) � = ∅ . A2. Let A i be bodies of evidence of Bel i ∈ M bel , i = 1 , 2, then Con ( Bel 1 , Bel 2 ) = 1 iff A ∩ B = ∅ for all A ∈ A 1 and B ∈ A 2 . A4. Let Bel i � Bel ′ i , i = 1 , 2, then Con ( Bel 1 , Bel 2 ) � Con ( Bel 1 , Bel 2 ); A6. Let Con ( Bel 1 , Bel 2 ) = a , where a ∈ [0 , 1] and Bel 1 , Bel 2 ∈ M bel , then there exist Bel ( k ) ∈ M bel , i, k = 1 , 2, such i that Bel i = (1 − a ) Bel (1) + aBel (2) i , i = 1 , 2, i Con ( Bel (1) 1 , Bel (1) 2 ) = 0, and Con ( Bel (2) 1 , Bel (2) 2 ) = 1. In addition, n � a) Con ( P 1 , P 2 ) = 1 − min { P 1 ( x i ) , P 2 ( x i ) } , P i ∈ M pr , i = 1 , 2; i =1 b) Con ( Bel 1 , Bel 2 ) = inf { Con ( P 1 , P 2 ) | P 1 ∈ P ( Bel 1 ) , P 2 ∈ P ( Bel 2 ) } . (HSE, Moscow, Russia) Conjunctive Rules Belief-2016 12 / 14
The axiomatics of Con ( Bel 1 , Bel 2 ) Lemma 3 Belief functions Bel 1 , Bel 2 ∈ M bel are absolutely contradictory, i.e. they obey the condition A2, iff there are disjoint sets A, B ∈ 2 X ( A ∩ B = ∅ ) such that Bel 1 ( A ) = Bel 2 ( B ) = 1. Lemma 4 Let a functional Φ : M pr × M pr → [0 , 1] obey axioms A1, A2 and A6. Then n Φ( P 1 , P 2 ) = 1 − � min { P 1 ( x i ) , P 2 ( x i ) } , P 1 , P 2 ∈ M pr . i =1 Theorem 1 Let a functional Φ : M bel × M bel → [0 , 1] obey axioms A1, A2, A4, and A6. Then it coincides with the contradiction measure Con on M bel × M bel . (HSE, Moscow, Russia) Conjunctive Rules Belief-2016 13 / 14
Thanks for you attention brone@mail.ru I.Rozenberg.gismps.ru (HSE, Moscow, Russia) Conjunctive Rules Belief-2016 14 / 14
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