Markov Logic Markov Logic Probability First-Order Logic - PDF document

Putting the Pieces Together Markov Logic Markov Logic Probability First-Order Logic Propositional Logic Markov Logic Definition  A Markov Logic Network (MLN) is a set of  A logical KB is a set of hard constraints pairs (F, w) where on the set of possible worlds  F is a formula in first-order logic  Let’s make them soft constraints :  w is a real number When a world violates a formula,  Together with a set of constants, It becomes less probable, not impossible it defines a Markov network with  Give each formula a weight  One node for each grounding of each predicate (Higher weight ⇒ Stronger constraint) in the MLN  One feature for each grounding of each formula F ( ) P(world) exp weights of formulas it satisfies � � in the MLN, with the corresponding weight w 1

Example: Friends & Smokers Example: Friends & Smokers x Smokes ( x ) Cancer ( x ) Smoking causes cancer. � � x , y Friends ( x , y ) ( Smokes ( x ) Smokes ( y ) ) Friends have similar smoking habits. � � � Example: Friends & Smokers Example: Friends & Smokers 1 . 5 x Smokes ( x ) Cancer ( x ) 1 . 5 x Smokes ( x ) Cancer ( x ) � � � � 1 . 1 x , y Friends ( x , y ) ( Smokes ( x ) Smokes ( y ) ) 1 . 1 x , y Friends ( x , y ) ( Smokes ( x ) Smokes ( y ) ) � � � � � � Two constants: Anna (A) and Bob (B) 2

Example: Friends & Smokers Example: Friends & Smokers 1 . 5 x Smokes ( x ) Cancer ( x ) 1 . 5 x Smokes ( x ) Cancer ( x ) � � � � 1 . 1 x , y Friends ( x , y ) ( Smokes ( x ) Smokes ( y ) ) 1 . 1 x , y Friends ( x , y ) ( Smokes ( x ) Smokes ( y ) ) � � � � � � Two constants: Anna (A) and Bob (B) Two constants: Anna (A) and Bob (B) Friends(A,B) Smokes(A) Smokes(B) Friends(A,A) Smokes(A) Smokes(B) Friends(B,B) Cancer(A) Cancer(B) Cancer(A) Cancer(B) Friends(B,A) Example: Friends & Smokers Example: Friends & Smokers 1 . 5 x Smokes ( x ) Cancer ( x ) 1 . 5 x Smokes ( x ) Cancer ( x ) � � � � 1 . 1 x , y Friends ( x , y ) ( Smokes ( x ) Smokes ( y ) ) 1 . 1 x , y Friends ( x , y ) ( Smokes ( x ) Smokes ( y ) ) � � � � � � Two constants: Anna (A) and Bob (B) Two constants: Anna (A) and Bob (B) Friends(A,B) Friends(A,B) Friends(A,A) Smokes(A) Smokes(B) Friends(B,B) Friends(A,A) Smokes(A) Smokes(B) Friends(B,B) Cancer(A) Cancer(B) Cancer(A) Cancer(B) Friends(B,A) Friends(B,A) 3

Markov Logic Networks Relation to Statistical Models  MLN is template for ground Markov nets  Special cases:  Obtained by making all  Probability of a world x : predicates zero-arity  Markov networks 1  Markov random fields � � P ( x ) exp w n ( x ) � = � �  Bayesian networks i i  Markov logic allows Z � � i  Log-linear models objects to be  Exponential models interdependent Weight of formula i No. of true groundings of formula i in x  Max. entropy models (non-i.i.d.)  Gibbs distributions  Typed variables and constants greatly reduce  Boltzmann machines size of ground Markov net  Discrete distributions  Logistic regression  Functions, existential quantifiers, etc.  Hidden Markov models  Conditional random fields  Open question: Infinite domains Relation to First-Order Logic MAP/MPE Inference  Infinite weights ⇒ First-order logic  Problem: Find most likely state of world given evidence  Satisfiable KB, positive weights ⇒ Satisfying assignments = Modes of distribution max P ( y | x )  Markov logic allows contradictions between y formulas Query Evidence 4

MAP/MPE Inference MAP/MPE Inference  Problem: Find most likely state of world  Problem: Find most likely state of world given evidence given evidence 1 � � � max w n ( x , y ) � max exp w n ( x , y ) � � i i i i y Z y i � � i x The MaxWalkSAT Algorithm MAP/MPE Inference for i ← 1 to max-tries do  Problem: Find most likely state of world solution = random truth assignment given evidence for j ← 1 to max-flips do if ∑ weights(sat. clauses) > threshold then max � w n ( x , y ) return solution i i y c ← random unsatisfied clause i with probability p  This is just the weighted MaxSAT problem flip a random variable in c  Use weighted SAT solver else flip variable in c that maximizes (e.g., MaxWalkSAT [Kautz et al., 1997] ) ∑ weights(sat. clauses)  Potentially faster than logical inference (!) return failure, best solution found 5

But … Memory Explosion Computing Probabilities  Problem:  P(Formula|MLN,C) = ? If there are n constants  MCMC: Sample worlds, check formula holds and the highest clause arity is c ,  P(Formula1|Formula2,MLN,C) = ? c the ground network requires O(n ) memory  If Formula2 = Conjunction of ground atoms  First construct min subset of network necessary  Solution: to answer query (generalization of KBMC) Exploit sparseness; ground clauses lazily  Then apply MCMC (or other) → LazySAT algorithm [Singla & Domingos, 2006]  Can also do lifted inference [Braz et al, 2005] Ground Network Construction But … Insufficient for Logic network ← Ø  Problem: queue ← query nodes Deterministic dependencies break MCMC repeat Near-deterministic ones make it very slow node ← front( queue ) remove node from queue  Solution: add node to network Combine MCMC and WalkSAT if node not in evidence then → MC-SAT algorithm [Poon & Domingos, 2006] add neighbors( node ) to queue until queue = Ø 6