On the Merit of Selecting Different Belief Merging Operators Pilar - PowerPoint PPT Presentation

On the Merit of Selecting Different Belief Merging Operators Pilar Pozos-Parra (joint work with Kevin McAreavey and Weiru Liu) University of Tabasco/Queen’s University of Belfast maripozos@gmail.com 5 July 2013 P. Pozos-Parra (UJAT) Selecting Belief Merging Operators 5 July 2013 1 / 27

Tabasco P. Pozos-Parra (UJAT) Selecting Belief Merging Operators 5 July 2013 2 / 27

Overview Motivation 1 Methods found in the literature 2 Our proposal of comparison 3 P. Pozos-Parra (UJAT) Selecting Belief Merging Operators 5 July 2013 3 / 27

Belief Merging Belief Bases (KBs) express beliefs, goals, etc. Profiles are multisets (bags) of KBs, E = { K 1 , K 2 , . . . , K m } . KBs in a profile may be inconsistent when taken together. Example: [Revesz 1993, Konieczny&Pino-P´ erez 1998] E = {{ ( s ∨ o ) ∧ ¬ d } , { ( ¬ s ∧ d ∧ ¬ o ) ∨ ( ¬ s ∧ ¬ d ∧ o ) } , { s ∧ d ∧ o }} � E ⊢ d ∧ ¬ d Merging is to combine these KBs into one, consistent KB, e.g. ∆( E ) = s ∧ ¬ d ∧ o . A merging operator ∆ is a function from profiles to KBs. P. Pozos-Parra (UJAT) Selecting Belief Merging Operators 5 July 2013 4 / 27

Some ∆’s KBs must be consistent Let E = { K 1 , K 2 , . . . , K m } , ∆ Max based on d Max ( E , v ) = max { d ( K 1 , v ) , . . . , d ( K m , v ) } ∆ Σ based on d Σ ( E , v ) = � m i =1 d ( K i , v ) ∆ Gmax based on d Gmax ( E , v ) = sort d ( d ( K 1 , v ) , . . . , d ( K m , v )) P. Pozos-Parra (UJAT) Selecting Belief Merging Operators 5 July 2013 5 / 27

Result of ∆’s E = { K 1 , K 2 , K 3 } , K 1 = { ( s ∨ o ) ∧ ¬ d } , K 2 = { ( ¬ s ∧ d ∧ ¬ o ) ∨ ( ¬ s ∧ ¬ d ∧ o ) } and K 3 = { s ∧ d ∧ o } . mod ( K 1 ) = { (1 , 0 , 0) , (0 , 0 , 1) , (1 , 0 , 1) } , mod ( K 2 ) = { (0 , 1 , 0) , (0 , 0 , 1) } and mod ( K 1 ) = { (1 , 1 , 1) } . ∆ max = ( s ∧ d ∧ ¬ o ) ∨ ( s ∧ ¬ d ∧ o ) ∨ ( ¬ s ∧ d ∧ o ), ∆ Σ = ( s ∧ ¬ d ∧ o ) ∨ ( ¬ s ∧ ¬ d ∧ o ), ∆ Gmax = ( s ∧ ¬ d ∧ o ) w d ( , K 1 ) ( d , K 2 ) ( d , K 3 ) d Max d Σ d Gmax (1 , 1 , 1) 1 2 0 2 3 (2 , 1 , 0) (1 , 1 , 0) 1 1 1 1 3 (1 , 1 , 1) (1 , 0 , 1) 0 1 1 1 2 (1 , 1 , 0) (1 , 0 , 0) 0 2 2 2 4 (2 , 2 , 0) (0 , 1 , 1) 1 1 1 1 3 (1 , 1 , 1) (0 , 1 , 0) 2 0 2 2 4 (2 , 2 , 2) (0 , 0 , 1) 0 0 2 2 2 (2 , 0 , 0) (0 , 0 , 0) 1 1 3 3 5 (3 , 1 , 1) P. Pozos-Parra (UJAT) Selecting Belief Merging Operators 5 July 2013 6 / 27

Application of ∆’s to Nutrition guide Nutrients required (age, gender, weight, height, level of physical activity) Aliments preferred by the user The ”Nutrition Facts” Recipes Postulates of the good alimentation (complete, sufficient, equilibrated, varied, innocuous) P. Pozos-Parra (UJAT) Selecting Belief Merging Operators 5 July 2013 7 / 27

Application of ∆’s to Nutrition guide E = { Nutrients required Aliments preferred by the user The ”Nutrition Facts” (the amounts of nutrients of aliments) Recipes (composed by aliments) Postulates of the good alimentation } � E ⊢ nutrients ∧ ¬ nutrients P. Pozos-Parra (UJAT) Selecting Belief Merging Operators 5 July 2013 8 / 27

Application of ∆’s to Cancer Diagnosis E = 171 cases : Variables or factors ( v 1 , v 2 , ..., v 8 ) Diagnosis ( d ) E training = 114 cases ∆( E training )= KB using [Pozos-Parra, Perrussel and Trevenin 2011] E testing = 57 cases without Diagnosis Obtaining diagnosis to the 57 cases using KB Table : Belief Merging Results for Different Factors #Fac TP FP FN TN Sensit Acc Specif PPV NPV AUC 8 27 15 1 14 96.43 71.93 48.28 64.29 93.33 0.723 7 28 12 0 17 100 78.95 58.62 70 100 0.793 6 28 18 0 11 100 68.42 37.93 60.87 100 0.69 5 28 29 0 0 100 49.12 0 49.12 CBD 0.5 4 28 29 0 0 100 49.12 0 49.12 CBD 0.5 [Abdul Kareem, Pozos-Parra and Wilson (on revision)] P. Pozos-Parra (UJAT) Selecting Belief Merging Operators 5 July 2013 9 / 27

Implementations of ∆’s For real life applications we need implementations of ∆’s We know only 2: Gorogiannis and Hunter 2008 1 Pozos-Parra, Perrussel and Trevenin 2011 2 Question Is there a method to compare the results? P. Pozos-Parra (UJAT) Selecting Belief Merging Operators 5 July 2013 10 / 27

Overview Motivation 1 Methods found in the literature 2 Our proposal of comparison 3 P. Pozos-Parra (UJAT) Selecting Belief Merging Operators 5 July 2013 11 / 27

Postulates There are postulates to characterize the process of belief merging. Definition Let E , E 1 , E 2 be belief profiles, K 1 and K 2 be consistent belief bases. Let ∆ be an operator which assigns to each belief profile E a belief base ∆( E ). ∆ is a merging operator if and only if it satisfies the following postulates: (M1) ∆( E ) is consistent (M2) if � E is consistent then∆( E ) ≡ � E (M3) if E 1 ≡ E 2 , then ∆( E 1 ) ≡ ∆( E 2 ) (M4) ∆( { K 1 , K 2 } ) ∧ K 1 is consistent if and only if ∆( { K 1 , K 2 } ) ∧ K 2 is consistent (M5) ∆( E 1 ) ∧ ∆( E 2 ) | = ∆( E 1 ⊔ E 2 ) (M6) if ∆( E 1 ) ∧ ∆( E 2 ) is consistent, then ∆( E 1 ⊔ E 2 ) | = ∆( E 1 ) ∧ ∆( E 2 ) [Konieczny&Pino-P´ erez 1998] P. Pozos-Parra (UJAT) Selecting Belief Merging Operators 5 July 2013 12 / 27

Comparing ∆’s Definition (Conformity relation) An operator ∆ 1 is more conforming than an operator ∆ 2 , denoted ∆ 1 ≥ ∆ 2 , if ∆ 1 satisfies more postulates than ∆ 2 . P. Pozos-Parra (UJAT) Selecting Belief Merging Operators 5 July 2013 13 / 27

Base satisfaction index Definition (weak drastic index) � 1 if K ∧ ∆( E ) is consistent , i w ( K , ∆( E )) = 0 otherwise . Definition (strong drastic index) � 1 if ∆( E ) | = K , i s ( K , ∆( E )) = 0 otherwise . [Everaere, Konieczny and Marquis 2007] P. Pozos-Parra (UJAT) Selecting Belief Merging Operators 5 July 2013 14 / 27

Base satisfaction index Definition (probabilistic index) This index takes the value of the probability of getting a model of K among the models of ∆( E ), formally: � 0 if | mod (∆( E )) | = 0 i p ( K , ∆( E )) = | mod ( K ) ∩ mod (∆( E )) | otherwise . | mod (∆( E )) | Definition (Dalal index) This index grows antimonotonically with the Hamming distance between the two bases under consideration, i.e., the minimal distance between a model of the base K and a model of base ∆( E ), formally: min w ∈ mod ( K ) , w ′∈ mod (∆( E )) d ( w , w ′ ) i d ( K , ∆( E )) = 1 − | P | . [Everaere, Konieczny and Marquis 2007] P. Pozos-Parra (UJAT) Selecting Belief Merging Operators 5 July 2013 15 / 27

Overview Motivation 1 Methods found in the literature 2 Our proposal of comparison 3 P. Pozos-Parra (UJAT) Selecting Belief Merging Operators 5 July 2013 16 / 27

Degrees of satisfaction Definition (Degree of satisfaction of belief bases) Function SAT : L × L → [0 , 1] is called a the degree of satisfaction of belief bases iff for any belief base K and K ′ , it satisfies the following postulates: Reflexivity: SAT ( K , K ′ ) = 1 iff mod ( K ′ ) ∩ mod ( K ) � = ∅ . Monotonicity: SAT ( K , K ′ ) ≥ SAT ( K , K ∗ ) iff mod ( K ′ ) ⊆ mod ( K ∗ ). Definition (Degree of satisfaction of belief profiles) Let E be a profile, SAT be a degree of satisfaction of belief bases and a be an aggregation function. The degree of satisfaction of E by K ′ based on SAT and a , denoted SAT a ( E , K ′ ), is defined as follows: SAT a ( E , K ′ ) = a K ∈ E SAT ( K , K ′ ) . P. Pozos-Parra (UJAT) Selecting Belief Merging Operators 5 July 2013 17 / 27

Max-min satisfaction Definition (The maximum degree of satisfaction of belief profiles) Let E be a profile and K ′ be a belief base, we say that SAT max ( E , K ′ ) is the maximum degree of satisfaction of E by K ′ iff SAT max ( E , K ′ ) = max K ∈ E SAT ( K , K ′ ) Definition (The minimum degree of satisfaction of belief profiles) Let E be a profile and K ′ be a belief base, we say that SAT min ( E , K ′ ) is the minimum degree of satisfaction of E by K ′ iff SAT min ( E , K ′ ) = min K ∈ E SAT ( K , K ′ ) P. Pozos-Parra (UJAT) Selecting Belief Merging Operators 5 July 2013 18 / 27

Relations over ∆’s Definition (Fairness relation) ∆ 1 is fairer than ∆ 2 , denoted ∆ 1 � ∆ 2 iff for all E , SAT max ( E , ∆ 1 ( E )) − SAT min ( E , ∆ 1 ( E )) ≤ SAT max ( E , ∆ 2 ( E )) − SAT min ( E , ∆ 2 ( E )) Definition (Satisfaction relation) ∆ 1 is more satisfactory than ∆ 2 , denoted ∆ 1 ⊒ ∆ 2 if for all E , SAT a ( E , ∆ 2 ( E )) ≤ SAT a ( E , ∆ 1 ( E )) Definition (Strength relation) ∆ 1 is stronger than ∆ 2 , denoted ∆ 1 ⊇ ∆ 2 , if for all E , mod (∆ 1 ( E )) ⊆ mod (∆ 2 ( E )) P. Pozos-Parra (UJAT) Selecting Belief Merging Operators 5 July 2013 19 / 27

Comparing ∆’s ∆ MCS ∆ Max ∆ Σ ∆ Gmax ∆ ⊤ ∆ MCS ≥ , � , ⊒ , ⊇ n/a �≥ �≥ �� ∆ Max ≥ ≥ , � , ⊒ , ⊇ �≥ �≥ , ⊒ ≥ ∆ Σ ≥ ≥ ≥ , � , ⊒ , ⊇ ≥ ≥ ∆ Gmax ≥ ≥ , ⊇ ≥ ≥ , � , ⊒ , ⊇ ≥ ∆ ⊤ ≥ , � , ⊒ , �⊇ ≥ , � , ⊒ , �⊇ �≥ , � , ⊒ , �⊇ �≥ , � , ⊒ , �⊇ ≥ , � , ⊒ , Table : Comparison of operators in terms of conformity, fairness, satisfaction and strength relations where n/a means not comparable or not found. P. Pozos-Parra (UJAT) Selecting Belief Merging Operators 5 July 2013 20 / 27