Markov Logic

Reviews
Markov Logic Stanley Kok Dept. of Computer Science & Eng. University of Washington Joint work with Pedro Domingos, Daniel Lowd, Hoifung Poon, Matt Richardson, Parag Singla and Jue Wang 1 Overview        Motivation Background Markov logic Inference Learning Software Applications 2 Motivation  Most learners assume i.i.d. data (independent and identically distributed)   One type of object Objects have no relation to each other  Real applications: dependent, variously distributed data   Multiple types of objects Relations between objects 3 Examples          Web search Medical diagnosis Computational biology Social networks Information extraction Natural language processing Perception Ubiquitous computing Etc. 4 Costs/Benefits of Markov Logic  Benefits    Better predictive accuracy Better understanding of domains Growth path for machine learning  Costs    Learning is much harder Inference becomes a crucial issue Greater complexity for user 5 Overview        Motivation Background Markov logic Inference Learning Software Applications 6 Markov Networks  Undirected graphical models Smoking Asthma  Cancer Cough Potential functions defined over cliques 1 P( x)    c ( xc ) Z c Z    c ( xc ) x c Smoking Cancer False False True True Ф(S,C) 4.5 4.5 2.7 4.5 7 False True False True Markov Networks  Undirected graphical models Smoking Asthma  Cancer Cough Log-linear model: 1   P( x)  exp   wi f i ( x)  Z  i  Weight of Feature i Feature i w1  1.5  1 if  Smoking  Cancer f1 (Smoking, Cancer )    0 otherwise 8 Hammersley-Clifford Theorem If Distribution is strictly positive (P(x) > 0) And Graph encodes conditional independences Then Distribution is product of potentials over cliques of graph Inverse is also true. (“Markov network = Gibbs distribution”) 9 Markov Nets vs. Bayes Nets Property Form Markov Nets Prod. potentials Bayes Nets Prod. potentials Potentials Cycles Indep. check Inference Arbitrary Allowed Cond. probabilities Forbidden Z=1 Some Partition func. Z = ? Indep. props. Some Graph separation D-separation MCMC, BP, etc. Convert to Markov 10 First-Order Logic      Constants, variables, functions, predicates E.g.: Anna, x, MotherOf(x), Friends(x, y) Literal: Predicate or its negation Clause: Disjunction of literals Grounding: Replace all variables by constants E.g.: Friends (Anna, Bob) World (model, interpretation): Assignment of truth values to all ground predicates 11 Overview        Motivation Background Markov logic Inference Learning Software Applications 12 Markov Logic: Intuition    P(world) exp weights of formulasit satisfies 13 A logical KB is a set of hard constraints on 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) Markov Logic: 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 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 14 Example: Friends & Smokers Smoking causes cancer. Friends have similar smoking habits. 15 Example: Friends & Smokers x Smokes( x )  Cancer( x ) x, y Friends( x, y )  Smokes( x )  Smokes( y )  16 Example: Friends & Smokers 1.5 x Smokes( x )  Cancer( x ) 1.1 x, y Friends( x, y )  Smokes( x )  Smokes( y )  17 Example: Friends & Smokers 1.5 x Smokes( x )  Cancer( x ) 1.1 x, y Friends( x, y )  Smokes( x )  Smokes( y )  Two constants: Anna (A) and Bob (B) 18 Example: Friends & Smokers 1.5 x Smokes( x )  Cancer( x ) 1.1 x, y Friends( x, y )  Smokes( x )  Smokes( y )  Two constants: Anna (A) and Bob (B) Smokes(A) Smokes(B) Cancer(A) Cancer(B) 19 Example: Friends & Smokers 1.5 x Smokes( x )  Cancer( x ) 1.1 x, y Friends( x, y )  Smokes( x )  Smokes( y )  Two constants: Anna (A) and Bob (B) Friends(A,B) Friends(A,A) Smokes(A) Smokes(B) Friends(B,B) Cancer(A) Cancer(B) Friends(B,A) 20 Example: Friends & Smokers 1.5 x Smokes( x )  Cancer( x ) 1.1 x, y Friends( x, y )  Smokes( x )  Smokes( y )  Two constants: Anna (A) and Bob (B) Friends(A,B) Friends(A,A) Smokes(A) Smokes(B) Friends(B,B) Cancer(A) Cancer(B) Friends(B,A) 21 Example: Friends & Smokers 1.5 x Smokes( x )  Cancer( x ) 1.1 x, y Friends( x, y )  Smokes( x )  Smokes( y )  Two constants: Anna (A) and Bob (B) Friends(A,B) Friends(A,A) Smokes(A) Smokes(B) Friends(B,B) Cancer(A) Cancer(B) Friends(B,A) 22 Markov Logic Networks   MLN is template for ground Markov nets Probability of a world x: 1   P( x)  exp   wi ni ( x)  Z  i  Weight of formula i No. of true groundings of formula i in x    Typed variables and constants greatly reduce size of ground Markov net Functions, existential quantifiers, etc. Infinite and continuous domains 23 Relation to Statistical Models  Special cases:             Markov networks Markov random fields Bayesian networks Log-linear models Exponential models Max. entropy models Gibbs distributions Boltzmann machines Logistic regression Hidden Markov models Conditional random fields Obtained by making all predicates zero-arity  Markov logic allows objects to be interdependent (non-i.i.d.) 24 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 25 Overview        Motivation Background Markov logic Inference Learning Software Applications 26 MAP/MPE Inference  Problem: Find most likely state of world given evidence max P( y | x) y Query Evidence 27 MAP/MPE Inference  Problem: Find most likely state of world given evidence 1   max exp   wi ni ( x, y)  y Zx  i  28 MAP/MPE Inference  Problem: Find most likely state of world given evidence max y  w n ( x, y ) i i i 29 MAP/MPE Inference  Problem: Find most likely state of world given evidence max y    w n ( x, y ) i i i  This is just the weighted MaxSAT problem Use weighted SAT solver (e.g., MaxWalkSAT [Kautz et al., 1997] ) Potentially faster than logical inference (!) 30 The WalkSAT Algorithm for i ← 1 to max-tries do solution = random truth assignment for j ← 1 to max-flips do if all clauses satisfied then return solution c ← random unsatisfied clause with probability p flip a random variable in c else flip variable in c that maximizes number of satisfied clauses return failure 31 The MaxWalkSAT Algorithm for i ← 1 to max-tries do solution = random truth assignment for j ← 1 to max-flips do if ∑ weights(sat. clauses) > threshold then return solution c ← random unsatisfied clause with probability p flip a random variable in c else flip variable in c that maximizes ∑ weights(sat. clauses) return failure, best solution found 32 But … Memory Explosion  Problem: If there are n constants and the highest clause arity is c, c the ground network requires O(n ) memory  Solution: Exploit sparseness; ground clauses lazily → LazySAT algorithm [Singla & Domingos, 2006] 33 Computing Probabilities     P(Formula|MLN,C) = ? MCMC: Sample worlds, check formula holds P(Formula1|Formula2,MLN,C) = ? If Formula2 = Conjunction of ground atoms   First construct min subset of network necessary to answer query (generalization of KBMC) Then apply MCMC (or other)  Can also do lifted inference [Braz et al, 2005] 34 Ground Network Construction network ← Ø queue ← query nodes repeat node ← front(queue) remove node from queue add node to network if node not in evidence then add neighbors(node) to queue until queue = Ø 35 MCMC: Gibbs Sampling state ← random truth assignment for i ← 1 to num-samples do for each variable x sample x according to P(x|neighbors(x)) state ← state with new value of x P(F) ← fraction of states in which F is true 36 But … Insufficient for Logic  Problem: Deterministic dependencies break MCMC Near-deterministic ones make it very slow  Solution: Combine MCMC and WalkSAT → MC-SAT algorithm [Poon & Domingos, 2006] 37 Overview        Motivation Background Markov logic Inference Learning Software Applications 38 Learning     Data is a relational database Closed world assumption (if not: EM) Learning parameters (weights) Learning structure (formulas) 39 Generative Weight Learning    Maximize likelihood Numerical optimization (gradient or 2nd order) No local maxima  log Pw ( x)  ni ( x)  Ew ni ( x) wi No. of times clause i is true in data Expected no. times clause i is true according to MLN  Requires inference at each step (slow!) 40 Pseudo-Likelihood PL ( x)   P ( xi | neighbors ( xi )) i     Likelihood of each variable given its neighbors in the data Does not require inference at each step Widely used in vision, spatial statistics, etc. But PL parameters may not work well for long inference chains 41 Discriminative Weight Learning  Maximize conditional likelihood of query (y) given evidence (x)  log Pw ( y | x)  ni ( x, y )  Ew ni ( x, y ) wi No. of true groundings of clause i in data Expected no. true groundings of clause i according to MLN  Approximate expected counts with:   counts in MAP state of y given x (with MaxWalkSAT) with MC-SAT 42 Structure Learning         Generalizes feature induction in Markov nets Any inductive logic programming approach can be used, but . . . Goal is to induce any clauses, not just Horn Evaluation function should be likelihood Requires learning weights for each candidate Turns out not to be bottleneck Bottleneck is counting clause groundings Solution: Subsampling 43 Structure Learning     Initial state: Unit clauses or hand-coded KB Operators: Add/remove literal, flip sign Evaluation function: Pseudo-likelihood + Structure prior Search: Beam, shortest-first, bottom-up [Kok & Domingos, 2005; Mihalkova & Mooney, 2007] 44 Overview        Motivation Background Markov logic Inference Learning Software Applications 45 Alchemy Open-source software including:  Full first-order logic syntax  Generative & discriminative weight learning  Structure learning  Weighted satisfiability and MCMC  Programming language features alchemy.cs.washington.edu 46 Overview        Motivation Background Markov logic Inference Learning Software Applications 47 Applications        Basics Logistic regression Hypertext classification Information retrieval Entity resolution Bayesian networks Etc. 48 Running Alchemy  Programs    MLN file    Infer Learnwts Learnstruct   Types (optional) Predicates Formulas  Options Database files 49 Uniform Distribn.: Empty MLN Example: Unbiased coin flips Type: flip = { 1, … , 20 } Predicate: Heads(flip) 1 Z 0 1 P( Heads( f ))  1  1 0 2 e Ze Z e0 50 Binomial Distribn.: Unit Clause Example: Biased coin flips Type: flip = { 1, … , 20 } Predicate: Heads(flip) Formula: Heads(f) Weight: Log odds of heads: 1 Z w  p  w  log  1 p     1 P(Heads(f))  1  p 1 0 w e  Z e 1 e Z By default, MLN includes unit clauses for all predicates (captures marginal distributions, etc.) 51 ew Multinomial Distribution Example: Throwing die throw = { 1, … , 20 } face = { 1, … , 6 } Predicate: Outcome(throw,face) Formulas: Outcome(t,f) ^ f != f’ => !Outcome(t,f’). Exist f Outcome(t,f). Types: Too cumbersome! 52 Multinomial Distrib.: ! Notation Example: Throwing die throw = { 1, … , 20 } face = { 1, … , 6 } Predicate: Outcome(throw,face!) Types: Formulas: Semantics: Arguments without “!” determine arguments with “!”. Also makes inference more efficient (triggers blocking). 53 Multinomial Distrib.: + Notation Example: Throwing biased die throw = { 1, … , 20 } face = { 1, … , 6 } Predicate: Outcome(throw,face!) Formulas: Outcome(t,+f) Types: Semantics: Learn weight for each grounding of args with “+”. 54 Logistic Regression  P(C  1 | F  f )  Logistic regression: log  P(C  0 | F  f )   a  bi f i    Type: obj = { 1, ... , n } Query predicate: C(obj) Evidence predicates: Fi(obj) Formulas: a C(x) bi Fi(x) ^ C(x) Resulting distribution: P(C  c, F  f )   expa   bi f i    P(C  1 | F  f )     a   bi f i Therefore: log   P(C  0 | F  f )   log    exp(0)     Alternative form: Fi(x) => C(x) 55 1   exp  ac   bi f i c  Z i   Text Classification page = { 1, … , n } word = { … } topic = { … } Topic(page,topic!) HasWord(page,word) !Topic(p,t) HasWord(p,+w) => Topic(p,+t) 56 Text Classification Topic(page,topic!) HasWord(page,word) HasWord(p,+w) => Topic(p,+t) 57 Hypertext Classification Topic(page,topic!) HasWord(page,word) Links(page,page) HasWord(p,+w) => Topic(p,+t) Topic(p,t) ^ Links(p,p') => Topic(p',t) Cf. S. Chakrabarti, B. Dom & P. Indyk, “Hypertext Classification Using Hyperlinks,” in Proc. SIGMOD-1998. 58 Information Retrieval InQuery(word) HasWord(page,word) Relevant(page) InQuery(+w) ^ HasWord(p,+w) => Relevant(p) Relevant(p) ^ Links(p,p’) => Relevant(p’) Cf. L. Page, S. Brin, R. Motwani & T. Winograd, “The PageRank Citation Ranking: Bringing Order to the Web,” Tech. Rept., Stanford University, 1998. 59 Entity Resolution Problem: Given database, find duplicate records HasToken(token,field,record) SameField(field,record,record) SameRecord(record,record) HasToken(+t,+f,r) ^ HasToken(+t,+f,r’) => SameField(+f,r,r’) SameField(f,r,r’) => SameRecord(r,r’) SameRecord(r,r’) ^ SameRecord(r’,r”) => SameRecord(r,r”) Cf. A. McCallum & B. Wellner, “Conditional Models of Identity Uncertainty with Application to Noun Coreference,” in Adv. NIPS 17, 2005. 60 Entity Resolution Can also resolve fields: HasToken(token,field,record) SameField(field,record,record) SameRecord(record,record) HasToken(+t,+f,r) ^ HasToken(+t,+f,r’) => SameField(f,r,r’) SameField(f,r,r’) <=> SameRecord(r,r’) SameRecord(r,r’) ^ SameRecord(r’,r”) => SameRecord(r,r”) SameField(f,r,r’) ^ SameField(f,r’,r”) => SameField(f,r,r”) More: P. Singla & P. Domingos, “Entity Resolution with Markov Logic”, in Proc. ICDM-2006. 61 Bayesian Networks    Use all binary predicates with same first argument (the object x). One predicate for each variable A: A(x,v!) One conjunction for each line in the CPT   A literal of state of child and each parent Weight = log P(Child|Parents)   Context-specific independence: One conjunction for each path in the decision tree Logistic regression: As before 62 Practical Tips     Add all unit clauses (the default) Implications vs. conjunctions Open/closed world assumptions Controlling complexity    Low clause arities Low numbers of constants Short inference chains   Use the simplest MLN that works Cycle: Add/delete formulas, learn and test 63 Summary   Most domains are non-i.i.d. Markov logic combines first-order logic and probabilistic graphical models   Syntax: First-order logic + Weights Semantics: Templates for Markov networks    Inference: LazySAT + MC-SAT Learning: LazySAT + MC-SAT + ILP + PL Software: Alchemy http://alchemy.cs.washington.edu 64

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