A betting metaphor for belief functions on MV-algebras and fuzzy - PowerPoint PPT Presentation

A betting metaphor for belief functions on MV-algebras and fuzzy epistemic states Tommaso Flaminio 1 Lluis Godo 2 M ANY V AL 2013 1 DiSTA, University of Insubria, Italy. tommaso.flaminio@uninsubria.it 2 IIIA - CSIC, Campus de la UAB, Spain. godo@iiia.csic.es

The extension problem: classical setting Two players, Bookmaker ( B ) and Gambler ( G ), play the following game: ◮ B fixes a finite class of events e 1 , . . . , e k and a Book α : e i �→ α i ∈ [ 0 , 1 ] ; ◮ G chooses stakes σ 1 , . . . , σ k in R one for each event e i and G pays to B the amount of � k i = 1 σ i · α i euros. ◮ In a future possible word V , for each e i , B pays to G : ◮ 0 euros if e i is false in V ; ◮ σ i euros if e i turns out to be true in V . ◮ Hence G and B are betting on unknown events and on the fact that they will turn out to be true. ◮ The total balance of the game for B is hence: k k k � � � σ i · α i − σ i · V ( e i ) = σ i · ( α i − V ( e i )) . i = 1 i = 1 i = 1 The book α is said to be a Dutch-Book provided that Gambler G has a strategy of bets ensuring her a sure win in every possible world V .

Formalization of the problem Let X = { V 1 , V 2 , . . . , V n } be a finite set of possible worlds, and let e 1 , . . . , e k in 2 X . A book is a map α : e i �→ α i ∈ [ 0 , 1 ] . Then α is coherent iff for every σ 1 , . . . , σ k ∈ R , there exists a possible world (i.e. a Boolean homomorphism) V j : 2 X → { 0 , 1 } such that k � σ i ( α ( e i ) − V j ( e i )) ≥ 0 . i = 1 By de Finetti’s theorem the coherence of α is equivalent to the existence of a probability measure P α on 2 X such that for each i , P α ( e i ) = α ( e i ) = α i .

For every possible world V j ∈ { V 1 , . . . , V n } let p j = � V j ( e 1 ) , . . . , V j ( e k ) � ∈ { 0 , 1 } k and let H = co { p j : j ∈ { 1 , 2 , . . . , n }} ⊆ [ 0 , 1 ] k . Then the book α is coherent (i.e. it extends to P α ) iff � α 1 , . . . , α k � ∈ H .

The case of many-valued events MV-algebras are the equivalent algebraic semantics for Łukasiewicz logic. These algebras are systems A = ( A , ⊕ , ¬ , 0 , 1 ) of type ( 2 , 1 , 0 , 0 ) . The class of MV-algebras forms a variety MV . (1) The typical example of MV-algebra is [ 0 , 1 ] MV = ([ 0 , 1 ] , ⊕ , ¬ , 0 , 1 ) where, for each x , y ∈ [ 0 , 1 ] , x ⊕ y = min { 1 , x + y } and ¬ x = 1 − x . The algebra [ 0 , 1 ] MV is generic for MV . (2) The class of all functions from [ 0 , 1 ] k to [ 0 , 1 ] which are continuous, piecewise linear with integer coefficients, together with operations ⊕ and ¬ defined as in [ 0 , 1 ] MV pointwise, is the free MV-algebra with k generators.

De Finetti’s coherence criterion can be stated in the frame of MV-algebras as follows (cf. Paris (7) and Mundici (6)): Let A be an MV-algebra, and let e 1 , . . . , e k be events in A . Let further α : e i �→ α i ∈ [ 0 , 1 ] be a book on the events e i ’s published by the bookmaker. Then α is coherent provided that for every choice of stakes σ 1 , . . . , σ k ∈ R , there exists a many-valued possible world V : A → [ 0 , 1 ] MV (i.e. an MV-homomorphism) such that k k k � � � σ i · α ( e i ) − σ i · V ( e i ) = σ i ( α ( e i ) − V ( e i )) ≥ 0 . i = 1 i = 1 i = 1

A state on an MV-algebra A is a map s : A → [ 0 , 1 ] such that: ◮ s ( 1 ) = 1; ◮ Whenever x ⊙ y = 0, s ( x ⊕ y ) = s ( x ) + s ( y ) , (where x ⊙ y = ¬ ( ¬ x ⊕ ¬ y ) ). Mundici (6) (and K¨ uhr-Mundici (5)) proved the following generalization of de Finetti’s theorem: Theorem. Let A be an MV-algebra, { e 1 , . . . , e k } ⊆ A , and α : e i �→ α i ∈ [ 0 , 1 ] . Then the following are equivalent: ◮ α is coherent; ◮ There exists a state s : A → [ 0 , 1 ] such that s ( e i ) = α i for each i = 1 , . . . , k ; ◮ There are MV-homomorphisms V 1 , . . . , V k + 1 : A → [ 0 , 1 ] MV such that � α 1 , . . . , α k � ∈ co { p j | j = 1 , . . . , k + 1 } . where p j = � V j ( e 1 ) , . . . , V j ( e k ) � ∈ [ 0 , 1 ] k .

Belief functions on Boolean algebras Belief functions on Boolean algebras can be introduced as follows: Let 2 X be a Boolean algebra of sets. For every A ⊆ X , consider the map � 1 if B ⊆ A β A : B ⊆ X �→ 0 otherwise. Then bel : 2 X → [ 0 , 1 ] is a belief function on 2 X provided that there exists a probability X measure P : 2 2 → [ 0 , 1 ] such that, for every A ∈ 2 X , bel ( A ) = P ( β A ) . A characterization of coherence in terms of extendability to a belief function was proved by Jaffray, 1989 (4). We will provide a similar result to the case of many-valued events.

Belief functions on MV-algebras of fuzzy sets In order to generalize belief function to MV-algebras of the form [ 0 , 1 ] X (with X finite), consider, for every a ∈ [ 0 , 1 ] X , the map ρ a so defined: ρ a : π ∈ [ 0 , 1 ] X �→ inf {¬ π ( x ) ⊕ a ( x ) : x ∈ X } . Notice that the map ρ a generalizes β A : for every A ∈ 2 X , the restriction of ρ A to 2 X coincides with β A . The MV-algebra R X generated by all the functions ρ a (for a ∈ [ 0 , 1 ] X ) is a separating MV-algebra of continuous functions. The MV-algebra R X is an MV-subalgebra of [ 0 , 1 ] [ 0 , 1 ] X . Definition. A map b : [ 0 , 1 ] X → [ 0 , 1 ] is belief function if there exists a state s : R X → [ 0 , 1 ] such that, for every a ∈ [ 0 , 1 ] X , b ( a ) = s ( ρ a ) . A belief function b is said to be normalized provided that b ( 0 ) = 0.

The map ρ ( · ) ( π ) For each π ∈ [ 0 , 1 ] X , the map N π : a ∈ [ 0 , 1 ] X �→ ρ a ( π ) ∈ [ 0 , 1 ] is a homogeneous necessity measure, Moreover N π ( · ) is normalized provided that there exists an x ∈ X such that π ( x ) = 1. Lemma. (1) The class of all necessity measures on [ 0 , 1 ] X coincides with the class { ρ ( · ) ( π ) : a ∈ [ 0 , 1 ] X �→ ρ a ( π ) | π ∈ [ 0 , 1 ] X } . (2) The class of all normalized necessity measures on [ 0 , 1 ] X coincides with the class { ρ ( · ) ( π ) : a ∈ [ 0 , 1 ] X �→ ρ a ( π ) | π ∈ [ 0 , 1 ] X , max x ∈ X π ( x ) = 1 } . ∗ Remark. In order to define (normalized) belief functions on [ 0 , 1 ] X we need two kind of mappings: ◮ A (normalized) necessity measure (equivalently a (normalized) possibility distribution); ◮ A state.

Idempotent (tropical) convex combinations Fix p 1 , . . . , p n ∈ [ 0 , 1 ] k . A point x ∈ [ 0 , 1 ] k is a bounded (normalized) min-plus convex combination of p 1 , . . . , p n if there exist λ 1 , . . . λ n ∈ [ − 1 , 0 ] (with � i ≤ n λ i = 0) such that � x ( j ) = ( λ i + p i ( j )) , for every j = 1 , . . . , k . i ≤ n The bounded min-plus convex hull of { p 1 , . . . , p n } is denoted bmp-co ( p 1 , . . . , p n ) , The bounded normalized min-plus convex hull of { p 1 , . . . , p n } is denoted nmp-co ( p 1 , . . . , p n ) , p 3 p 3 p 2 p 2 p 1 p 1 O O

Theorem. [F -Godo, (3)] Let e 1 , . . . , e k ∈ [ 0 , 1 ] X , and let α : e i �→ α i be an assignment. Then the following hold: 1. α extends to a belief function b on [ 0 , 1 ] X iff there are MV-homomorphisms V x : [ 0 , 1 ] X → [ 0 , 1 ] MV (for x ∈ X ) such that � α 1 , . . . , α k � ∈ co ( bmp-co ( { p x : x ∈ X } )) . 2. α extends to a normalized belief function b on [ 0 , 1 ] X iff there are MV-homomorphisms V x : [ 0 , 1 ] X → [ 0 , 1 ] MV (for x ∈ X ) such that � α 1 , . . . , α s � ∈ co ( nmp-co ( { p x : x ∈ X } )) . (For every x ∈ X , p x = � V x ( e 1 ) , . . . , V x ( e k ) � )

Let X = { V 1 , V 2 , V 3 } , and let e 1 , e 2 ∈ [ 0 , 1 ] X be: e 1 = � 1 / 2 , 5 / 6 , 1 / 5 � and e 2 = � 1 / 3 , 1 / 2 , 9 / 10 � , and the following assignments α 1 ( e 1 ) = 1 / 3 , α 1 ( e 2 ) = 2 / 5 (1) and α 2 ( e 1 ) = 2 / 3 , α 2 ( e 2 ) = 18 / 40 (2) The events e 1 and e 2 corresponds, in [ 0 , 1 ] 2 , to the points: p 1 = � V 1 ( e 1 ) , V 1 ( e 2 ) � = � 1 / 2 , 1 / 3 � p 2 = � V 2 ( e 1 ) , V 2 ( e 2 ) � = � 5 / 6 , 1 / 2 � p 3 = � V 3 ( e 1 ) , V 3 ( e 2 ) � = � 1 / 5 , 9 / 10 �

Extending to a Normalized Belief Function p 3 = ( 1 / 5 , 9 / 10 ) p 2 = ( 5 / 6 , 1 / 2 ) α 2 α 1 p 1 = ( 1 / 2 , 1 / 3 ) O

Extending to a Normalized Necessity Measure p 3 = ( 1 / 5 , 9 / 10 ) p 2 = ( 5 / 6 , 1 / 2 ) α 2 α 2 α 1 α 1 p 1 = ( 1 / 2 , 1 / 3 ) O

Extending to a Normalized Belief Function p 3 = ( 1 / 5 , 9 / 10 ) p 2 = ( 5 / 6 , 1 / 2 ) α 2 α 2 α 1 α 1 p 1 = ( 1 / 2 , 1 / 3 ) O

Towards a betting interpretation Turning back to the previous result, given a finite class of events in [ 0 , 1 ] X , and a book α : e i �→ α i , the following are equivalent: ◮ There exists a (normalized) belief function b : [ 0 , 1 ] X → [ 0 , 1 ] such that b ( e i ) = α i for each i ; ◮ There exists a state s : R X → [ 0 , 1 ] such that, for each i = 1 , . . . , k s ( ρ e i ) = α i . ◮ The book α R : ρ e i �→ α i is coherent (in terms of states ), i.e. for every stakes σ 1 , . . . , σ k ∈ R , there exists a MV-homomorphism V : R X → [ 0 , 1 ] MV (i.e. a MV-possible world) such that k � σ i ( α ( ρ e i ) − V ( ρ e i )) ≥ 0 . i = 1