markov logic: combining logic and probability parag singla dept. of computer science &...
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Markov Logic: Combining Logic and Probability
Parag SinglaDept. of Computer Science & Engineering
Indian Institute of Technology Delhi
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Overview
Motivation & Background Markov logic Inference & Learning Abductive Plan Recognition
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Social Network and Smoking Behavior
Smoking Cancer
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Social Network and Smoking Behavior
Smoking leads to Cancer
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Social Network and Smoking Behavior
Smoking leads to Cancer
Friendship Similar Smoking Habits
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Social Network and Smoking Behavior
Smoking leads to Cancer
Friendship leads to Similar Smoking Habits
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Statistical Relational AI
Real world problems characterized by Entities and Relationships Uncertain Behavior
Relational Models Horn clauses, SQL queries, first-order logic
Statistical Models Markov networks, Bayesian networks
How to combine the two? Markov Logic
Markov Networks + First Order Logic
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Statistical Relational AI Probabilistic logic [Nilsson, 1986] Statistics and beliefs [Halpern, 1990] Knowledge-based model construction
[Wellman et al., 1992] Stochastic logic programs [Muggleton, 1996] Probabilistic relational models [Friedman et al., 1999] Bayesian Logic Programs
[Kersting and De Raedt 2001] Relational Markov networks [Taskar et al., 2002] BLOG [Milch et al., 2005] Markov logic [Richardson & Domingos, 2006]
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First-Order Logic
Constants, variables, functions, predicates Anil, x, MotherOf(x), Friends(x,y)
Grounding: Replace all variables by constants Friends (Anna, Bob)
Formula: Predicates connected by operators Smokes(x) Cancer(x)
Knowledge Base (KB): A set of formulas Can be equivalently converted into a clausal form
World: Assignment of truth values to all ground atoms
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First-Order Logic
Deal with finite first-order logic Assumptions
Unique Names Domain Closure Known Functions
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Markov Networks Undirected graphical models
Log-linear model:
Weight of Feature i Feature i
otherweise0
Cancer Smokingif1)CancerSmoking,(1f
5.11 w
Cancer
CoughAsthma
Smoking
iii xfw
ZxP )(exp
1)(
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Overview
Motivation & Background Markov logic Inference & Learning Abductive Plan Recognition
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Markov Logic [Richardson & Domingos 06]
A logical KB is a set of hard constraintson the set of possible worlds
Let’s make them soft constraints:When a world violates a formula,It becomes less probable, not impossible
Give each formula a weight(Higher weight Stronger constraint)
satisfiesit formulas of weightsexpP(world)
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Definition
A Markov Logic Network (MLN) is a set of pairs (F, w) where F is a formula in first-order logic w is a real number
Together with a finite set of constants,it defines a Markov network with One node for each grounding of each predicate in
the MLN One feature for each grounding of each formula F
in the MLN, with the corresponding weight w
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Example: Friends & Smokers
habits. smoking similar have Friends
cancer. causes Smoking
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Example: Friends & Smokers
)()(),(,
)()(
ySmokesxSmokesyxFriendsyx
xCancerxSmokesx
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Example: Friends & Smokers
)()(),(,
)()(
ySmokesxSmokesyxFriendsyx
xCancerxSmokesx
1.1
5.1
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Example: Friends & Smokers
Two constants: Anil (A) and Bunty (B)
)()(),(,
)()(
ySmokesxSmokesyxFriendsyx
xCancerxSmokesx
1.1
5.1
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Example: Friends & Smokers
Cancer(A)
Smokes(A) Smokes(B)
Cancer(B)
Two constants: Anil (A) and Bunty (B)
)()(),(,
)()(
ySmokesxSmokesyxFriendsyx
xCancerxSmokesx
1.1
5.1
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Example: Friends & Smokers
Cancer(A)
Smokes(A)Friends(A,A)
Friends(B,A)
Smokes(B)
Friends(A,B)
Cancer(B)
Friends(B,B)
Two constants: Anil (A) and Bunty (B)
)()(),(,
)()(
ySmokesxSmokesyxFriendsyx
xCancerxSmokesx
1.1
5.1
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Example: Friends & Smokers
Cancer(A)
Smokes(A)Friends(A,A)
Friends(B,A)
Smokes(B)
Friends(A,B)
Cancer(B)
Friends(B,B)
Two constants: Anil (A) and Bunty (B)
)()(),(,
)()(
ySmokesxSmokesyxFriendsyx
xCancerxSmokesx
1.1
5.1
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Example: Friends & Smokers
Cancer(A)
Smokes(A)Friends(A,A)
Friends(B,A)
Smokes(B)
Friends(A,B)
Cancer(B)
Friends(B,B)
Two constants: Anil (A) and Bunty (B)
)()(),(,
)()(
ySmokesxSmokesyxFriendsyx
xCancerxSmokesx
1.1
5.1
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Example: Friends & Smokers
Cancer(A)
Smokes(A)Friends(A,A)
Friends(B,A)
Smokes(B)
Friends(A,B)
Cancer(B)
Friends(B,B)
Two constants: Anil (A) and Bunty (B)
)()(),(,
)()(
ySmokesxSmokesyxFriendsyx
xCancerxSmokesx
1.1
5.1
State of the World {0,1} Assignment to the nodes
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Markov Logic Networks
MLN is template for ground Markov nets
Probability of a world x:
formulasgroundkkk xfw
ZxP
)(exp1
)(
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Markov Logic Networks
MLN is template for ground Markov nets
Probability of a world x:
Weight of formula i No. of true groundings of formula i in x
formulas MLN
)(exp1
)(i
ii xnwZ
xP
formulasgroundkkk xfw
ZxP
)(exp1
)(
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Relation to Statistical Models
Special cases: Markov networks Markov random fields Bayesian networks Log-linear models Exponential models Logistic regression Hidden Markov models Conditional random fields
Obtained by making all predicates zero-arity
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Relation to First-Order Logic
Infinite weights First-order logic Satisfiable KB, positive weights
Satisfying assignments = Modes of distribution Markov logic allows contradictions between
formulas Relaxing Assumptions
Known Functions (Markov Logic in Infinite Domains) [Singla & Domingos 07]
Unique Names (Entity Resolution with Markov Logic) [Singla & Domingos 06]
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Overview
Motivation & Background Markov logic Inference & Learning Abductive Plan Recognition
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Inference
Cancer(A)
Smokes(A)?Friends(A,A)
Friends(B,A)
Smokes(B)?
Friends(A,B)
Friends(B,B)
Cancer(B)
blue ? – non-evidence (unknown)green/orange – evidence (known)
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MPE Inference Problem: Find most likely state of world
given evidence
iii
x
yxnwZ
xyP ),(exp1
)|(
Query Evidence
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MPE Inference Problem: Find most likely state of world
given evidence
i
iixy
yxnwZ
),(exp1
maxarg
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MPE Inference
Problem: Find most likely state of world given evidence
i
iiy
yxnw ),(maxarg
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MPE Inference
Problem: Find most likely state of world given evidence
This is just the weighted MaxSAT problem Use weighted SAT solver
(e.g., MaxWalkSAT [Kautz et al. 97] )
i
iiy
yxnw ),(maxarg
Lazy Grounding of Clauses: LazySAT [Singla & Domingos 06]
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Marginal Inference
Problem: Find the probability of query atoms given evidence
iii
x
yxnwZ
xyP ),(exp1
)|(
Query Evidence
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Marginal Inference
Problem: Find the probability of query atoms given evidence
iii
x
yxnwZ
xyP ),(exp1
)|(
Query Evidence
Computing Zx takes exponential time!
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Marginal Inference
Problem: Find the probability of query atoms given evidence
iii
x
yxnwZ
xyP ),(exp1
)|(
Query Evidence
Approximate Inference: Gibbs Sampling, Message Passing[Richardson & Domingos 06, Poon & Domingos 06, Singla & Domingos 08]
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Learning Parameters
)()(),(,
)()(
ySmokesxSmokesyxFriendsyx
xCancerxSmokesx
?w
?w
2
1
Three constants: Anil, Bunty, Chaya
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Learning Parameters
)()(),(,
)()(
ySmokesxSmokesyxFriendsyx
xCancerxSmokesx
?w
?w
2
1
Smokes
Smokes(Anil)
Smokes(Bunty)
Closed World Assumption: Anything not in the database is assumed false.
Three constants: Anil, Bunty, Chaya
Cancer
Cancer(Anil)
Cancer(Bunty)
Friends
Friends(Anil, Bunty)
Friends(Bunty, Anil)
Friends(Anil, Chaya)
Friends(Chaya, Anil)
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Learning Parameters
)()(),(,
)()(
ySmokesxSmokesyxFriendsyx
xCancerxSmokesx
?w
?w
2
1
Smokes
Smokes(Anil)
Smokes(Bunty)
Three constants: Anil, Bunty, Chaya
Cancer
Cancer(Anil)
Cancer(Bunty)
Friends
Friends(Anil, Bunty)
Friends(Bunty, Anil)
Friends(Anil, Chaya)
Friends(Chaya, Anil)
Maximize the Likelihood: Use Gradient Based Approaches [Singla & Domingos 05, Lowd & Domingos 07]
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Learning Structure
Smokes
Smokes(Anil)
Smokes(Bunty)
Cancer
Cancer(Anil)
Cancer(Bunty)
Friends
Friends(Anil, Bob)
Friends(Bunty, Anil)
Friends(Anil, Chaya)
Friends(Chaya, Anil)
Three constants: Anil, Bunty, Chaya
Can we learn the set of the formulas in the MLN?
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Learning Structure
)()(),(,
)()(
ySmokesxSmokesyxFriendsyx
xCancerxSmokesx
?w
?w
2
1
Can we refine the set of the formulas in the MLN?
Smokes
Smokes(Anil)
Smokes(Bunty)
Cancer
Cancer(Anil)
Cancer(Bunty)
Friends
Friends(Anil, Bob)
Friends(Bunty, Anil)
Friends(Anil, Chaya)
Friends(Chaya, Anil)
Three constants: Anil, Bunty, Chaya
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Learning Structure
)()(),(,
)()(
ySmokesxSmokesyxFriendsyx
xCancerxSmokesx
?w
?w
2
1
Can we refine the set of the formulas in the MLN?
Smokes
Smokes(Anil)
Smokes(Bunty)
Cancer
Cancer(Anil)
Cancer(Bunty)
Friends
Friends(Anil, Bob)
Friends(Bunty, Anil)
Friends(Anil, Chaya)
Friends(Chaya, Anil)
Three constants: Anil, Bunty, Chaya
),(),(, xyFriendsyxFriendsyx ?w3
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Learning Structure
ILP style search for formuals[Kok & Domingos 05, 07, 09, 10]
Smokes
Smokes(Anil)
Smokes(Bunty)
Cancer
Cancer(Anil)
Cancer(Bunty)
Friends
Friends(Anil, Bob)
Friends(Bunty, Anil)
Friends(Anil, Chaya)
Friends(Chaya, Anil)
Three constants: Anil, Bunty, Chaya
)()(),(,
)()(
ySmokesxSmokesyxFriendsyx
xCancerxSmokesx
?w
?w
2
1
),(),(, xyFriendsyxFriendsyx ?w3
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Alchemy
Open-source software including: Full first-order logic syntax Inference algorithms Parameter & structure learning algorithms
alchemy.cs.washington.edu
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Overview
Motivation & Background Markov logic Inference & Learning Abductive Plan Recognition
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Applications
Web-mining Collective Classification Link Prediction Information retrieval Entity resolution Activity Recognition Image Segmentation & De-noising
Social Network Analysis Computational Biology Natural Language
Processing Robot mapping Abductive Plan
Recognition More..
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Abduction
Abduction: Given the observations and the background, find the best explanation
Given: Background knowledge (B) A set of observations (O)
To Find: A hypothesis, H, a set of assumptions
B H , B H O
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Plan Recognition
Given planning knowledge and a set of low-level actions, identify the top level plan
Involves abductive reasoning
B: Planning Knowledge (Background)
O: Set of low-level Actions (Observations)
H: Top Level Plan (Hypothesis)
B H , B H | O
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Plan Recognition Example
Emergency Response Domain [Blaylock & Allen 05]
Background Knowledge heavy_snow(loc) drive_hazard(loc) block_road(loc)
accident(loc) clear_wreck(crew,loc) block_road(loc)
Observation block_road(Plaza)
Possible Explanations Heavy Snow? Accident?
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Abduction using Markov logic Given heavy_snow(loc) drive_hazard(loc) block_road(loc)
accdent(loc) clear_wreck(crew, loc) block_road(loc)
Observation: block_road(plaza)
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Abduction using Markov logic
Given heavy_snow(loc) drive_hazard(loc) block_road(loc)
accdent(loc) clear_wreck(crew, loc) block_road(loc)
Observation: block_road(plaza)
Does not work!
Rules are true independent of antecedents Need to go from effect to cause
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Introducing Hidden Cause
heavy_snow(loc) drive_hazard(loc) block_road(loc)
heavy_snow(loc) drive_hazard(loc) rb_C1(loc)
rb_C1(loc) Hidden Cause
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Introducing Hidden Cause
heavy_snow(loc) drive_hazard(loc) block_road(loc)
heavy_snow(loc) drive_hazard(loc) rb_C1(loc)
rb_C1(loc) Hidden Cause
rb_C1(loc) block_road(loc)
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Introducing Hidden Cause
heavy_snow(loc) drive_hazard(loc) block_road(loc)
heavy_snow(loc) drive_hazard(loc) rb_C1(loc)
rb_C1(loc) Hidden Cause
rb_C1(loc) block_road(loc)
accident(loc) clear_wreck(loc, crew) block_road(loc)
accident(loc) clear_wreck(loc) rb_C2(loc, crew)
rb_C2(loc, crew)
rb_C2(loc,crew) block_road(loc)
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Introducing Reverse Implication
block_road(loc) rb_C1(loc) v ( crew rb_C2(loc, crew))
Explanation 2: accident(loc) clear_wreck(loc) rb_C2(loc, crew)
Explanation 1: heavy_snow(loc) clear_wreck(loc) rb_C1(loc)
Multiple causes combined via reverse implication
Existential quantification
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Existential quantification
Low-Prior on Hidden Causes
block_road(loc) rb_C1(loc) v ( crew rb_C2(loc, crew))
Explanation 2: accident(loc) clear_wreck(loc) rb_C2(loc, crew)
Explanation 1: heavy_snow(loc) clear_wreck(loc) rb_C1(loc)
Multiple causes combined via reverse implication
-w1 rb_C1(loc)-w2 rb_C2(loc, crew)
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Hidden Causes: Avoiding Blow-up
drive_hazard(Plaza)
heavy_snow(Plaza)
accident(Plaza)
clear_wreck(Tcrew, Plaza)rb_C1
(Plaza)rb_C2
(Tcrew, Plaza)
block_road(Tcrew, Plaza)
Hidden Cause Model
Max clique size = 3
[Singla & Domingos 2011]
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Hidden Causes: Avoiding Blow-up
drive_hazard(Plaza)
heavy_snow(Plaza)
accident(Plaza)
clear_wreck(Tcrew, Plaza)
drive_hazard(Plaza)
heavy_snow(Plaza)
accident(Plaza)
clear_wreck(Tcrew, Plaza)rb_C1
(Plaza)rb_C2
(Tcrew, Plaza)
block_road(Tcrew, Plaza)
block_road(Tcrew, Plaza)
Pair-wise Constraints[Kate & Mooney 2009]
Max clique size = 5
Hidden Cause Model
Max clique size = 3
[Singla & Domingos 2011]
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Second Issue: Ground Network Too Big!
Grounding out the full network may be costly Many irrelevant nodes/clauses are created Complicates learning/inference Can focus the grounding (KBMC)
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Abductive Model Construction
block_road(Plaza)
heavy_snow(Plaza)
drive_hazard(Plaza)
Observation:block_road(Plaza)
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Abductive Model Construction
Constants:…, Mall, City_Square, ...
block_road(Mall)
heavy_snow(Mall)
drive_hazard(Mall)
block_road(City_Square)
drive_hazard(City_Square)
heavy_snow(City_Square)
block_road(Plaza)
heavy_snow(Plaza)
drive_hazard(Plaza)
Observation:block_road(Plaza)
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Abductive Model Construction
Constants:…, Mall, City_Square, ...
Not a part of abductive
proof trees!
block_road(City_Square)
drive_hazard(City_Square)
heavy_snow(City_Square)
block_road(Mall)
heavy_snow(Mall)
drive_hazard(Mall)
block_road(Plaza)
heavy_snow(Plaza)
drive_hazard(Plaza)
Observation:block_road(Plaza)
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Abductive Model Construction
Constants:…, Mall, City_Square, ...
Not a part of abductive
proof trees!
block_road(City_Square)
drive_hazard(City_Square)
heavy_snow(City_Square)
block_road(Mall)
heavy_snow(Mall)
drive_hazard(Mall)
block_road(Plaza)
heavy_snow(Plaza)
drive_hazard(Plaza)
Observation:block_road(Plaza)
Backward chaining to get proof trees [Stickel 1988]
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Abductive Markov Logic[Singla & Domingos 11]
Re-encode the MLN rules Introduce reverse implications
Construct ground Markov network Use abductive model construction
Perform learning and inference
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Summary
Real world applications Entities and Relations Uncertainty
Unifying logical and statistical AI Markov Logic – simple and powerful model Need to do to efficient learning and inference Applications: Abductive Plan Recognition