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/Queens University of Belfast maripozos@gmail.com 5 July 2013 P. Pozos-Parra (UJAT)
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
Selecting Belief Merging Operators 5 July 2013 1 / 27
Selecting Belief Merging Operators 5 July 2013 2 / 27
1
Motivation
2
Methods found in the literature
3
Our proposal of comparison
Selecting Belief Merging Operators 5 July 2013 3 / 27
Belief Bases (KBs) express beliefs, goals, etc. Profiles are multisets (bags) of KBs, E = {K1, K2, . . . , Km}. 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.
Selecting Belief Merging Operators 5 July 2013 4 / 27
KBs must be consistent Let E = {K1, K2, . . . , Km}, ∆Max based on dMax(E, v) = max{d(K1, v), . . . , d(Km, v)} ∆Σ based on dΣ(E, v) = m
i=1 d(Ki, v)
∆Gmax based on dGmax(E, v) = sortd(d(K1, v), . . . , d(Km, v))
Selecting Belief Merging Operators 5 July 2013 5 / 27
E = {K1, K2, K3}, K1 = {(s ∨ o) ∧ ¬d}, K2 = {(¬s ∧ d ∧ ¬o) ∨ (¬s ∧ ¬d ∧ o)} and K3 = {s ∧ d ∧ o}. mod(K1) = {(1, 0, 0), (0, 0, 1), (1, 0, 1)}, mod(K2) = {(0, 1, 0), (0, 0, 1)} and mod(K1) = {(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(, K1) (d, K2) (d, K3) dMax dΣ dGmax (1, 1, 1) 1 2 2 3 (2, 1, 0) (1, 1, 0) 1 1 1 1 3 (1, 1, 1) (1, 0, 1) 1 1 1 2 (1, 1, 0) (1, 0, 0) 2 2 2 4 (2, 2, 0) (0, 1, 1) 1 1 1 1 3 (1, 1, 1) (0, 1, 0) 2 2 2 4 (2, 2, 2) (0, 0, 1) 2 2 2 (2, 0, 0) (0, 0, 0) 1 1 3 3 5 (3, 1, 1)
Selecting Belief Merging Operators 5 July 2013 6 / 27
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)
Selecting Belief Merging Operators 5 July 2013 7 / 27
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
Selecting Belief Merging Operators 5 July 2013 8 / 27
E= 171 cases : Variables or factors (v1, v2, ..., v8) Diagnosis (d) Etraining = 114 cases ∆(Etraining)= KB using [Pozos-Parra, Perrussel and Trevenin 2011] Etesting = 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 17 100 78.95 58.62 70 100 0.793 6 28 18 11 100 68.42 37.93 60.87 100 0.69 5 28 29 100 49.12 49.12 CBD 0.5 4 28 29 100 49.12 49.12 CBD 0.5
[Abdul Kareem, Pozos-Parra and Wilson (on revision)]
Selecting Belief Merging Operators 5 July 2013 9 / 27
For real life applications we need implementations of ∆’s We know only 2:
1
Gorogiannis and Hunter 2008
2
Pozos-Parra, Perrussel and Trevenin 2011
Question
Is there a method to compare the results?
Selecting Belief Merging Operators 5 July 2013 10 / 27
1
Motivation
2
Methods found in the literature
3
Our proposal of comparison
Selecting Belief Merging Operators 5 July 2013 11 / 27
There are postulates to characterize the process of belief merging.
Definition
Let E, E1, E2 be belief profiles, K1 and K2 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 E1 ≡ E2, then ∆(E1) ≡ ∆(E2) (M4) ∆({K1, K2}) ∧ K1 is consistent if and only if ∆({K1, K2}) ∧ K2 is consistent (M5) ∆(E1) ∧ ∆(E2) | = ∆(E1 ⊔ E2) (M6) if ∆(E1) ∧ ∆(E2) is consistent, then ∆(E1 ⊔ E2) | = ∆(E1) ∧ ∆(E2) [Konieczny&Pino-P´ erez 1998]
Selecting Belief Merging Operators 5 July 2013 12 / 27
Definition (Conformity relation)
An operator ∆1 is more conforming than an operator ∆2, denoted ∆1 ≥ ∆2, if ∆1 satisfies more postulates than ∆2.
Selecting Belief Merging Operators 5 July 2013 13 / 27
Definition (weak drastic index)
iw(K, ∆(E)) =
if K ∧ ∆(E) is consistent,
Definition (strong drastic index)
is(K, ∆(E)) =
if ∆(E) | = K,
[Everaere, Konieczny and Marquis 2007]
Selecting Belief Merging Operators 5 July 2013 14 / 27
Definition (probabilistic index)
This index takes the value of the probability of getting a model of K among the models of ∆(E), formally: ip(K, ∆(E)) =
|mod(K)∩mod(∆(E))| |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: id(K, ∆(E)) = 1 − minw∈mod(K),w′∈mod(∆(E)) d(w, w′) |P| . [Everaere, Konieczny and Marquis 2007]
Selecting Belief Merging Operators 5 July 2013 15 / 27
1
Motivation
2
Methods found in the literature
3
Our proposal of comparison
Selecting Belief Merging Operators 5 July 2013 16 / 27
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 SATa(E, K ′), is defined as follows: SATa(E, K ′) = aK∈ESAT(K, K ′).
Selecting Belief Merging Operators 5 July 2013 17 / 27
Definition (The maximum degree of satisfaction of belief profiles)
Let E be a profile and K ′ be a belief base, we say that SATmax(E, K ′) is the maximum degree of satisfaction of E by K ′ iff SATmax(E, K ′) = maxK∈ESAT(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 SATmin(E, K ′) is the minimum degree of satisfaction of E by K ′ iff SATmin(E, K ′) = minK∈ESAT(K, K ′)
Selecting Belief Merging Operators 5 July 2013 18 / 27
Definition (Fairness relation)
∆1 is fairer than ∆2, denoted ∆1 ∆2 iff for all E, SATmax(E, ∆1(E)) − SATmin(E, ∆1(E)) ≤ SATmax(E, ∆2(E)) − SATmin(E, ∆2(E))
Definition (Satisfaction relation)
∆1 is more satisfactory than ∆2, denoted ∆1 ⊒ ∆2 if for all E, SATa(E, ∆2(E)) ≤ SATa(E, ∆1(E))
Definition (Strength relation)
∆1 is stronger than ∆2, denoted ∆1 ⊇ ∆2, if for all E, mod(∆1(E)) ⊆ mod(∆2(E))
Selecting Belief Merging Operators 5 July 2013 19 / 27
∆MCS ∆Max ∆Σ ∆Gmax ∆⊤ ∆MCS ≥, ,⊒,⊇ n/a ≥ ≥
≥ ≥, ,⊒,⊇ ≥ ≥, ⊒ ≥ ∆Σ ≥ ≥ ≥, ,⊒,⊇ ≥ ≥ ∆Gmax ≥ ≥,⊇ ≥ ≥, ,⊒,⊇ ≥ ∆⊤ ≥ , , ⊒ , ⊇ ≥, , ⊒,⊇ ≥, , ⊒ ,⊇ ≥, , ⊒,⊇ ≥, ,⊒, Table : Comparison of operators in terms of conformity, fairness, satisfaction and strength relations where n/a means not comparable or not found.
Selecting Belief Merging Operators 5 July 2013 20 / 27
The inconsistency measure ILPm is defined as the normalized minimum number of inconsistent truth values in the LPm models of the belief base. Formally, let K be a belief base: ILPm(K) = minw∈modLP(K)(|w!|) |P|
Definition (Base-level inconsistency index)
The base-level inconsistency index is defined as: ii(K, ∆(E)) = 1 − I(K ∪ ∆(E)). The five base satisfaction indexes (is, iw and ip, id and iL) are degrees
Selecting Belief Merging Operators 5 July 2013 21 / 27
Profile is as satisfied as the satisfaction of its least satisfied element. imin(E, ∆(E)) = minK∈Ei(K, ∆(E)) Profile is satisfied holistically, as the sum of the satisfaction of its elements. iΣ(E, ∆(E)) = ΣK∈Ei(K, ∆(E)).
Selecting Belief Merging Operators 5 July 2013 22 / 27
Example
Let E = {K1, K2, K3} where K1 = {(S ∨ O) ∧ ¬D}, K2 = {(¬S ∧ D ∧ ¬O) ∨ (¬S ∧ ¬D ∧ O)} and K3 = {S ∧ D ∧ O}. Then ∆max = (s ∧ d ∧ ¬o) ∨ (s ∧ ¬d ∧ o) ∨ (¬s ∧ d ∧ o), ∆Σ = (s ∧ ¬d ∧ o) ∨ (¬s ∧ ¬d ∧ o), ∆Gmax = (s ∧ ¬d ∧ o)
Selecting Belief Merging Operators 5 July 2013 23 / 27
∆MCS(E) ∆Max(E) ∆Σ(E) ∆Gmax(E) iw(K1, ∆a(E)) 1 1 1 1 iw(K2, ∆a(E)) 1 1 iw(K3, ∆a(E)) 1 iw,min(E, ∆a(E)) 1 iw,Σ(E, ∆a(E)) 3 1 2 1 is(K1, ∆a(E)) 1 is(K2, ∆a(E)) 1 is(K3, ∆a(E)) is,min(E, ∆a(E)) is,Σ(E, ∆a(E)) 2 ip(K1, ∆a(E)) 0.5 0.33 1 1 ip(K2, ∆a(E)) 0.5 0.5 ip(K3, ∆a(E)) 0.5 ip,min(E, ∆a(E)) 0.5 ip,Σ(E, ∆a(E)) 1.5 0.33 1.5 1 id(K1, ∆a(E)) 1 1 1 1 id(K2, ∆a(E)) 1 0.66 1 0.66 id(K3, ∆a(E)) 1 0.66 0.66 0.66 id,min(E, ∆a(E)) 1 0.66 0.66 0.66 id,Σ(E, ∆a(E)) 3 2.33 2.66 2.33 iL(. . . , ∆a(E)) same as id
Selecting Belief Merging Operators 5 July 2013 24 / 27
input: E if ∆1(E) is stronger ∆2(E) then Ω = {w|w ∈ ∆2 but w ∈ ∆1} if ∆1(E) is fairer and more satisfactory than Ω then ∆1 gives a better result for E ∆Gmax is a better operator than ∆Max and ∆Σ for E = {K1, K2, K3} where K1 = {(S ∨ O) ∧ ¬D}, K2 = {(¬S ∧ D ∧ ¬O) ∨ (¬S ∧ ¬D ∧ O)} and K3 = {S ∧ D ∧ O}.
Selecting Belief Merging Operators 5 July 2013 25 / 27
Preliminary proposal of comparison Possible implementation To consider integrity constraints
Selecting Belief Merging Operators 5 July 2013 26 / 27
Selecting Belief Merging Operators 5 July 2013 27 / 27