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**Efficient Inference Methods for Probabilistic Logical Models**

Sriraam Natarajan Dept of Computer Science, University of Wisconsin-Madison

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**Take-Away Message Inference in SRL Models is very hard!!!!**

This talk – Presents 3 different yet related inference methods The methods are independent of the underlying formalism They have been applied to different kinds of problems

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**Propositional Model! The World is inherently Uncertain**

Graphical Models (here e.g. a Bayesian network) - Model uncertainty explicitly by representing the joint distribution Fever Ache Random Variables Influenza Direct Influences Propositional Model!

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**Real-World Data (Dramatically Simplified)**

Non- i.i.d PatientID Date Physician Symptoms Diagnosis P /1/ Smith palpitations hypoglycemic P /1/ Jones fever, aches influenza PatientID Gender Birthdate P M /22/63 Shared Parameters Solution: First-Order Logic / Relational Databases PatientID Date Lab Test Result PatientID SNP1 SNP2 … SNP500K P AA AB BB P AB BB AA P /1/01 blood glucose P /9/01 blood glucose Multi-Relational PatientID Date Prescribed Date Filled Physician Medication Dose Duration P /17/ /18/ Jones prilosec mg 3 months

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**Statistical Relational Learning (SRL)**

Logic + Probability = Probabilistic Logic aka Statistical Relational Learning Models Logic Statistical Relational Learning (SRL) Add Probabilities Probabilities Add Relations Uncertainty in SRL Models is captured by probabilities, weights or potential functions

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**Alphabetic Soup => Endless Possibilities**

Probabilistic Relational Models (PRM) Bayesian Logic Programs (BLP) PRISM Stochastic Logic Programs (SLP) Independent Choice Logic (ICL) Markov Logic Networks (MLN) Relational Markov Nets (RMN) CLP-BN Relational Bayes Nets (RBN) Probabilistic Logic Progam (PLP) ProbLog …. Web data (web) Biological data (bio) Social Network Analysis (soc) Bibliographic data (cite) Epidimiological data (epi) Communication data (comm) Customer networks (cust) Collaborative filtering problems (cf) Trust networks (trust) … Fall 2003– OSU, Spring 2004 UW, Spring Purdue, Fall 2008 – CMU

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**Key Problem - Inference**

Equivalent to counting 3SAT Models => #P-complete More pronounced in SRL Models Prohibitively large number of Objects and Relations Inference has been the biggest bottleneck for the use of SRL Models in practice

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**Grounding / Propositionalization**

Difficulty(C,D), Grade(S,C,G) :- Satisfaction(S) 1 student s1, 10 Courses Diff(c6,d3) Diff(c1,d1) Diff(c5,d1) Grade(s1,c5,A) Grade(s1,c6,B) Grade(s1,c1,B) Diff(c4,d4) Diff(c7,d2) Satisfaction(S) Grade(s1,c4,A) Grade(s1,c7,A) Diff(c8,d2) Diff(c3,d2) Grade(s1,c3,B) Grade(s1,c8,A) Grade(s1,c10,A) Diff(c2,d1) Diff(c9,d4) Grade(s1,c2,A) Diff(c10,d2) Grade(s1,c9,A)

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**Realistic Example – Gene-fold Prediction**

Thanks to Irene Ong

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**Recent Advances in SRL Inference**

Preprocessing for Inference FROG – Shavlik & Natarajan (2009) Lifted Exact Inference Lifted Variable Elimination – Poole (2003), Braz et al(2005) Milch et al (2008) Lifted VE + Aggregation – Kisynski & Poole (2009) Sampling Methods MCMC techniques – Milch & Russell (2006) Logical Particle Filter – Natarajan et al (2008), ZettleMoyer et al (2007) Lazy Inference – Poon et al (2008) Approximate Methods Lifted First-Order Belief Propagation – Singla & Domingos (2008) Counting Belief Propagation – Kersting et al (2009) MAP Inference – Riedel (2008) Bounds Propagation Anytime Belief Propagation – Braz et al (2009)

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**Fast Reduction of Grounded MLNs**

Counting Belief Propagation Anytime Lifted Belief Propagation Conclusion

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**Fast Reduction of Grounded MLNs**

Counting Belief Propagation Anytime Lifted Belief Propagation Conclusion

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**Markov Logic Networks Weighted logic**

Standard approach 1) Assume finite number of constants 2) Create all possible groundings 3) Perform statistical inference (often via sampling) Weight of formula i No. of true groundings of formula i in x (Richardson & Domingos, MLJ 2005)

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**Counting Satisfied Groundings**

Typically lots of redundancy in FOL sentences x, y, z p(x) ⋀ q(x, y, z) ⋀ r(z) w(x, y, z) If p(John) = false, then formula = true for all Y and Z values

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**Factoring Out the Evidence**

Let A = weighted sum of formula satisfied by evidence Let Bi = weighted sum of formula in world i not satisfied by evidence Prob(world i ) = e Bi e B … + e Bn e A + Bi e A + B … + e A + Bn

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Take-Away Message - I Efficiently factor out those formula groundings that evidence satisfies Can potentially eliminate the need for approximate inference

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**Worked Example 1012 The Evidence**

x, y, z GradStudent(x) ⋀ Prof(y) ⋀ Prof(z) ⋀ TA(x, z) ⋀ SameGroup(y, z) AdvisedBy(x, y) 10,000 People at some school 2000 Graduate students 1000 Professors 1000 TAs 500 Pairs of professors in the same group The Evidence Total Num of Groundings = |x| |y| |z| = 1012 1012

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**GradStudent(x) FROG keeps only these X values 2000 Grad Students 8000**

GradStudent(x) ⋀ Prof(y) ⋀ Prof(z) ⋀ TA(x,z) ⋀ SameGroup(y,z) AdvisedBy(x,y) FROG keeps only these X values GradStudent(P1) GradStudent(P3) … Grad Students GradStudent(x) True GradStudent(P1) ¬ GradStudent(P2) GradStudent(P3) … ¬ GradStudent(P2) ¬ GradStudent(P4) … 8000 Others False All these values for X satisfy the clause, regardless of Y and Z 1012 2 × 1011 Instead of 104 values for X, have 2 x 103

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**Prof(y) 1000 Professors 9000 Others 2 × 1011 2 × 1010**

GradStudent(x) ⋀ Prof(y) ⋀ Prof(z) ⋀ TA(x,z) ⋀ SameGroup(y,z) AdvisedBy(x,y) 1000 Professors Prof(P2) … Prof(y) True ¬ Prof(P1) Prof(P2) … 9000 Others False ¬ Prof(P1) … 2 × 1011 2 × 1010

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**<<< Same as Prof(y) >>>**

GradStudent(x) ⋀ Prof(y) ⋀ Prof(z) ⋀ TA(x,z) ⋀ SameGroup(y,z) AdvisedBy(x,y) <<< Same as Prof(y) >>> 2 × 1010 2 × 109

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**1000 true SameGroup’s SameGroup(y, z) 106 – 1000 Others 2 × 109**

GradStudent(x) ⋀ Prof(y) ⋀ Prof(z) ⋀ TA(x,z) ⋀ SameGroup(y,z) AdvisedBy(x,y) SameGroup(P1, P2) … 1000 true SameGroup’s SameGroup(y, z) True 106 Combinations 106 – Others ¬ SameGroup(P2, P5) … False 2000 values of X 1000 Y:Z combinations 2 × 109 2 × 106

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**GradStudent(x) ⋀ Prof(y) ⋀ Prof(z) ⋀ TA(x,z) ⋀ SameGroup(y,z) AdvisedBy(x,y)**

1000 TA’s TA(P7,P5) … TA(x, z) True 2 × 106 Combinations 2 × 106 – 1000 Others ¬ TA(P8,P4) … False ≤ values of X ≤ Y:Z combinations ≤ 106

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**Original number of groundings = 1012**

GradStudent(x) ⋀ Prof(y) ⋀ Prof(z) ⋀ TA(x,z) ⋀ SameGroup(y,z) AdvisedBy(x,y) 1012 Original number of groundings = 1012 106 Final number of groundings ≤ 106

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**Sample Results: UWash-CSE**

Fully Grounded Net FROG’s Reduced Net FROG’s Reduced Net without One Challenging Rule advisedBy(x,y) advisedBy(x,z) samePerson(y,z))

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**Fast Reduction of Grounded MLNs**

Counting Belief Propagation Anytime Lifted Belief Propagation Conclusion

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Belief Propagation Message passing algorithm – Inference on graphical models For factor graphs Exact – if the factor graph is a tree Approximate when it has cycles Loopy BP does not guarantee convergence, but is found to be very useful in practice X3 f1 X1 f2 X2

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Belief Propagation Identical Factors

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Take-Away Message – II Counting shared factors can result in great efficiency gains for (loopy) belief propagation

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**Counting Belief Propagation**

Two Steps Compress Factor Graph Run modified BP

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Step 1: Compression

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**Step 2: Modified Belief Propagation**

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**Factored Frontier (FF)**

Probabilistic inference over time is central to many AI problems In contrast to static domains, we need approximation Variables easily become correlated over time by virtue of sharing common influences in the past Factored Frontier [Murphy and Weiss 01] Unroll DBN Run (loopy) BP Lifted First-Order FF: Use CBP in place of BP

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**Lifted First-order Factored Frontier**

Successor fluent 20 people over 10 time steps Max number of friends 5 Cancer never observed Time step randomly selected

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**Fast Reduction of Grounded MLNs**

Counting Belief Propagation Anytime Lifted Belief Propagation Conclusion

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**The Need for Shattering**

Lifted BP depends on clusters of variables being symmetric, that is, sending and receiving identical messages In other words, it is about dividing random variables in cases – called as “shattering”

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**Intuition for Anytime Lifted BP**

in(House, Town) next(House,Another) earthquake(Town) Alarm can go off due to an earthquake lives(Another,Neighbor) alarm(House) saw(Neighbor,Someone) burglary(House) masked(Someone) A “prior” factor makes alarm going off unlikely without those causes Alarm can go off due to burglary partOf(Entrance,House) in(House,Item) broken(Entrance) missing(Item)

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**Intuition for Anytime Lifted BP**

alarm(House) earthquake(Town) in(House, Town) burglary(House) next(House,Another) lives(Another,Neighbor) saw(Neighbor,Someone) masked(Someone) in(House,Item) missing(Item) partOf(Entrance,House) broken(Entrance) Given a home in sf with home2 and home3 next to it with neighbors jim and mary, each seeing person1 and person2, several items in home, including a missing ring and non-missing cash, broken front but not broken back entrances to home, an earthquake in sf, what is the probability that home’s alarm goes off?

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**Lifted Belief Propagation**

Model for house ≠ home and town ≠ sf not shown Message passing over entire model before obtaining query answer next(home,home2) in(home, sf) Complete shattering before belief propagation starts lives(home2,jim) … earthquake(sf) saw(jim,person1) masked(person1) next(home,home3) alarm(home) lives(home2,mary) saw(mary,person2) burglary(home) masked(person2) in(home,cash) partOf(front,home) … missing(cash) broken(front) in(home,Item) in(home,ring) partOf(back,home) Item not in { ring,cash,…} … missing(ring) missing(Item) broken(back)

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**Intuition for Anytime Lifted BP**

next(home,home2) Evidence in(home, sf) lives(home2,jim) … earthquake(sf) saw(jim,person1) Given earthquake, we already have a good lower bound, regardless of burglary branch Query masked(person1) next(home,home3) alarm(home) lives(home2,mary) saw(mary,person2) burglary(home) Wasted shattering! Wasted shattering! Wasted shattering! Wasted shattering! Wasted shattering! masked(person2) in(home,cash) partOf(front,home) … missing(cash) broken(front) in(home,Item) in(home,ring) partOf(back,home) Item not in { ring,cash,…} … missing(ring) missing(Item) broken(back)

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**Using only a portion of a model**

By using only a portion, we don’t have to shatter other parts of the model How can we use only a portion? A solution for propositional models already exists: box propagation (Mooij & Kappen NIPS ‘08)

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Box Propagation A way of getting bounds on query without examining entire network. [0, 1] A

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Box Propagation A way of getting bounds on query without examining entire network. [0.36, 0.67] [0, 1] A B f1

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Box Propagation A way of getting bounds on query without examining entire network. [0,1] [0.1, 0.6] [0.38, 0.50] [0.05, 0.5] f2 ... A B f1 [0,1] f3 ... [0.32, 0.4]

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Box Propagation A way of getting bounds on query without examining entire network. [0.2,0.8] [0.3, 0.4] [0.41, 0.44] [0.17, 0.3] f2 ... A B f1 [0,1] f3 ... [0.32, 0.4]

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Box Propagation A way of getting bounds on query without examining entire network. 0.45 0.32 0.42 0.21 f2 ... A B f1 0.3 f3 ... 0.36 Convergence after all messages are collected

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**Take-Away Message - III**

Anytime BP = Incremental Shattering + Box Propagation

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**Anytime Lifted Belief Propagation**

Start from query alone [0,1] alarm(home) The algorithm works by picking a cluster variable and including the factors in its blanket

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**Anytime Lifted Belief Propagation**

in(home, Town) earthquake(Town) [0.1, 0.9] alarm(home) burglary(home) (alarm(home), in(home,Town), earthquake(Town)) after unifying alarm(home) and alarm(House) in (alarm(House), in(House,Town), earthquake(Town)) producing constraint House = home Again, through unification Blanket factors alone can determine a bound on query

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**Anytime Lifted Belief Propagation**

(in(home, sf)) in(home, sf) earthquake(sf) Cluster in(home, Town) unifies with in(home, sf) in (in(home, sf)) (which represents evidence) splitting cluster around Town = sf [0.1, 0.9] alarm(home) burglary(home) in(home, Town) Bound remains the same because we still haven’t considered evidence on earthquakes Town ≠ sf earthquake(Town)

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**Anytime Lifted Belief Propagation**

in(home, sf) (earthquake(sf)) represents the evidence that there was an earthquake earthquake(sf) [0.8, 0.9] alarm(home) burglary(home) Now query bound becomes narrow No need to further expand (and shatter) other branches If bound is good enough, there is no need to further expand (and shatter) other branches in(home, Town) Town ≠ sf earthquake(Town)

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**Anytime Lifted Belief Propagation**

in(home, sf) earthquake(sf) [0.85, 0.9] partOf(front,home) alarm(home) burglary(home) broken(front) in(home, Town) We can keep expanding at will for narrower bounds… Now query bound becomes narrow Town ≠ sf earthquake(Town)

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**Anytime Lifted Belief Propagation**

next(home,home2) in(home, sf) … until convergence, if desired. lives(home2,jim) … earthquake(sf) saw(jim,person1) 0.8725 masked(person1) next(home,home3) alarm(home) lives(home2,mary) saw(mary,person2) burglary(home) masked(person2) in(home,cash) partOf(front,home) … missing(cash) broken(front) in(home,Item) in(home,ring) partOf(back,home) Item not in { ring,cash,…} … missing(ring) missing(Item) broken(back)

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**Connection to Resolution Refutation**

Incremental shattering corresponds to building a proof tree in(home, sf) earthquake(sf) true alarm(home) burglary(home) … in(home,L), L not in {sf} earthquake(L), L not in {sf}

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**Fast Reduction of Grounded MLNs**

Counting Belief Propagation Anytime Lifted Belief Propagation Conclusion

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**Conclusion Inference is the key issue in several SRL formalisms**

FROG - Keeps the count of unsatisfied groundings Order of Magnitude reduction in number of groundings Compares favorably to Alchemy in different domains Counting BP - BP + grouping nodes sending and receiving identical messages Conceptually easy, scaleable BP algorithm Applications to challenging AI tasks Anytime BP – Incremental Shattering + Box Propagation Only the most necessary fraction of model considered and shattered Status – Implementation and evaluation

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**Conclusion Algorithms are independent of representation**

Variety of Applications Parameter Learning of Relational Models Social Networks Object Recognition Link Prediction Activity Recognition Model Counting Bio-Medical Applications Relational RL

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**Future Work FROG CBP Anytime BP SRL Models**

Combine with Lifted Inference Exploit commonality across rules CBP Integrate with Parameter Learning in SRL Models Extend to Multi-Agent RL, Lifted Pairwise BP Anytime BP Heuristic to expand the network Understand closer connections to Resolution SRL Models Learning Dynamic SRL Models Structure Learning remains an open issue

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**Acknowledgements* Babak Ahmadi - Fraunhofer Institute**

Rodrigo de Salvo Braz – SRI International Hung Bui – SRI International Vitor Santos Costa – U Porto Kristian Kersting - Fraunhofer Institute Gautam Kunapuli – UW Madison David Page – UW Madison Stuart Russell – UC Berkeley Jude Shavlik – UW Madison Prasad Tadepalli – Oregon State University * Ordered by Last name

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