A Short Introduction to Probabilistic Soft Logic Angelika Kimmig, Stephen H. Bach, Matthias Broecheler, Bert Huang and Lise Getoor NIPS Workshop on Probabilistic Programming 2012 http://psl.umiacs.umd.edu 1
Probabilistic Soft Logic (PSL) [Broecheler et al, UAI 10] 2
Probabilistic Soft Logic (PSL) [Broecheler et al, UAI 10] Declarative language to specify graphical models 2
Probabilistic Soft Logic (PSL) [Broecheler et al, UAI 10] Declarative language to specify graphical models • Logical atoms with soft truth values in [0,1] 2
Probabilistic Soft Logic (PSL) [Broecheler et al, UAI 10] Declarative language to specify graphical models • Logical atoms with soft truth values in [0,1] • Dependencies as weighted first order rules 2
Probabilistic Soft Logic (PSL) [Broecheler et al, UAI 10] Declarative language to specify graphical models • Logical atoms with soft truth values in [0,1] • Dependencies as weighted first order rules • Support for similarity functions and aggregation 2
Probabilistic Soft Logic (PSL) [Broecheler et al, UAI 10] Declarative language to specify graphical models • Logical atoms with soft truth values in [0,1] • Dependencies as weighted first order rules • Support for similarity functions and aggregation • Linear (in)equality constraints 2
Probabilistic Soft Logic (PSL) [Broecheler et al, UAI 10] Declarative language to specify graphical models • Logical atoms with soft truth values in [0,1] • Dependencies as weighted first order rules • Support for similarity functions and aggregation • Linear (in)equality constraints new approach NIPS 12 Efficient MPE inference: continuous convex optimization 2
Applications • Collective classification • Ontology alignment • Entity resolution • Link prediction • Trust in social networks • Social group modeling • Personalized medicine • ... 3
Ontology Alignment Organization provides work for interacts buys Service & Products Customers Employees develops sells to helps Developer Sales Person Staff Software Hardware IT Services Company works for develop buys interacts with Products & Services Customer Employee sells helps Technician Sales Accountant Software Dev Hardware Consulting 1 4
Ontology Alignment Organization provides work for similar interacts buys Service & Products Customers Employees names develops sells to helps Developer Sales Person Staff Software Hardware IT Services Company works for develop buys interacts with Products & Services Customer Employee sells helps Technician Sales Accountant Software Dev Hardware Consulting 1 4
Ontology Alignment similar ranges Organization provides work for similar interacts buys Service & Products Customers Employees names develops sells to helps Developer Sales Person Staff Software Hardware IT Services Company works for develop buys interacts with Products & Services Customer Employee sells helps Technician Sales Accountant Software Dev Hardware Consulting 1 4
Ontology Alignment similar ranges Organization provides work for similar interacts buys Service & Products Customers Employees names develops sells to helps Developer Sales Person Staff Software Hardware IT Services Company works for develop buys interacts with Products & Services Customer Employee sells helps Technician Sales Accountant Software Dev Hardware Consulting 1 similar subconcepts 4
Trust Modeling 0.2 0.5 carla emma dan 0.8 0.9 0.5 0.8 0.7 0.4 ann 0.9 fred bob 5 5
Trust Modeling 0.2 0.5 carla emma dan 0.8 0.9 0.5 0.8 0.7 0.4 ann 0.9 fred bob trusts(X,Y) ∧ trusts(Y,Z) → trusts(X,Z) 5 5
Trust Modeling 0.2 0.5 carla emma dan 0.8 0.9 0.5 0.8 0.7 ✔ 0.4 ann 0.9 fred bob trusts(X,Y) ∧ trusts(Y,Z) → trusts(X,Z) 5 5
Trust Modeling 0.2 0.5 carla emma dan ✗ 0.8 0.9 0.5 0.8 0.7 0.4 ann 0.9 fred bob trusts(X,Y) ∧ trusts(Y,Z) → trusts(X,Z) 5 5
Voter Opinion Modeling friend spouse carla emma dan colleague friend spouse friend friend colleague ann spouse fred bob 5 6
Voter Opinion Modeling friend spouse carla emma dan colleague friend spouse friend friend colleague ann spouse fred bob 5 6
Voter Opinion Modeling ? friend spouse carla emma dan colleague friend spouse friend ? friend colleague ann ? spouse fred bob 5 6
PSL Program ? friend spouse friend(carla,emma)=0.9 c e friend d colleague friend friend(bob,dan)=0.4 spouse friend spouse(ann,bob)=1.0 ? colleague spouse prefers(ann, )=0.8 ? a f ... b 7
PSL Program ? friend spouse friend(carla,emma)=0.9 c e friend d colleague friend friend(bob,dan)=0.4 spouse friend spouse(ann,bob)=1.0 ? colleague spouse prefers(ann, )=0.8 ? a f ... b 0.3: lives(A,S) ∧ majority(S,P) → prefers(A,P) 0.8: spouse(B,A) ∧ prefers(B,P) → prefers(A,P) 0.1: similarAge(B,A) ∧ prefers(B,P) → prefers(A,P) 0.4: prefers(A,P) → prefersAvg({A.friend},P) partial-functional: prefers 7
Constraints ? friend spouse c e friend d colleague friend spouse friend ? colleague spouse ? a f b partial-functional: prefers 8
Constraints ? friend spouse c e friend d colleague friend spouse friend ? colleague spouse ? a f b partial-functional: prefers prefers(A, )+prefers(A, ) ≤ 1.0 8
Local Rules ? friend spouse c e friend d colleague friend spouse friend ? colleague spouse ? a f b 0.3: lives(A,S) ∧ majority(S,P) → prefers(A,P) 9
Propagation Rules ? friend spouse c e friend d colleague friend spouse friend ? colleague spouse ? a f b 0.8: spouse(B,A) ∧ prefers(B,P) → prefers(A,P) 10
Similarity Rules ? friend spouse c e friend d colleague friend spouse friend ? colleague spouse ? a f b 0.1: similarAge(B,A) ∧ prefers(B,P) → prefers(A,P) Similarity function with range [0,1] 11
Sets ? friend spouse c e friend d colleague friend spouse friend ? colleague spouse ? a f b 0.4: prefers(A,P) → prefersAvg({A.friend},P) 12
Sets ? friend spouse c e friend d colleague friend spouse friend ? colleague spouse ? a f b all X such that friend(A,X) 0.4: prefers(A,P) → prefersAvg({A.friend},P) 12
Sets ? friend spouse c e friend d colleague friend spouse friend ? colleague spouse ? a f b all X such that friend(A,X) 0.4: prefers(A,P) → prefersAvg({A.friend},P) truth value ≔ average truth value of prefers(X,P) 12
PSL Program • Ground atoms = random variables • Soft truth value assignments • Assignment satisfying more rules more likely • Constraints to rule out unwanted assignments 13
Probabilistic Model 0 1 X X f ( I ) = 1 w r ( d g ( I )) k Z exp @ − A r ∈ P g ∈ G ( r ) 14
Probabilistic Model 0 1 X X f ( I ) = 1 w r ( d g ( I )) k Z exp @ − A r ∈ P g ∈ G ( r ) Interpretation 14
Probabilistic Model 0 1 X X f ( I ) = 1 w r ( d g ( I )) k Z exp @ − A r ∈ P g ∈ G ( r ) Interpretation Set of rule groundings 14
Probabilistic Model 0 1 X X f ( I ) = 1 w r ( d g ( I )) k Z exp @ − A r ∈ P g ∈ G ( r ) Rule’s weight Interpretation Set of rule groundings 14
Probabilistic Model Ground rule’s distance from satisfaction given I 0 1 X X f ( I ) = 1 w r ( d g ( I )) k Z exp @ − A r ∈ P g ∈ G ( r ) Rule’s weight Interpretation Set of rule groundings 14
Probabilistic Model Ground rule’s distance from satisfaction given I ∈ {1,2} 0 1 X X f ( I ) = 1 w r ( d g ( I )) k Z exp @ − A r ∈ P g ∈ G ( r ) Rule’s weight Interpretation Set of rule groundings 14
Probabilistic Model Ground rule’s distance from satisfaction given I ∈ {1,2} 0 1 X X f ( I ) = 1 w r ( d g ( I )) k Z exp @ − A r ∈ P g ∈ G ( r ) Rule’s weight Interpretation Set of rule groundings Normalization constant 0 1 Z X X w r ( d g ( J )) k Z = exp @ − A J ∈ I r ∈ P g ∈ G ( r ) 14
Distance from Satisfaction d r ( I ) = max { 0 , I ( body ) − I ( head ) } 15
⇔ Distance from Satisfaction d r ( I ) = max { 0 , I ( body ) − I ( head ) } body → head satisfied truth value of body ≤ truth value of head 15
⇔ Distance from Satisfaction d r ( I ) = max { 0 , I ( body ) − I ( head ) } body → head satisfied truth value of body ≤ truth value of head I ( v 1 ∧ v 2 ) = max { 0 , I ( v 1 ) + I ( v 2 ) − 1 } Lukasiewicz I ( v 1 ∨ v 2 ) = min { I ( v 1 ) + I ( v 2 ) , 1 } t-norm I ( ¬ l 1 ) = 1 − I ( v 1 ) 15
Distance from Satisfaction similarAge(bob,ann) ∧ prefers(bob, ) → prefers(ann, ) 0.8 0.5 0.5 16
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