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Markov Logic networks

A brief introduction to. Markov Logic networks. Markov Networks. Undirected graphical models. Smoking. Cancer. Asthma. Cough. Potential functions defined over cliques. Markov Networks. Undirected graphical models. Smoking. Cancer. Asthma. Cough. Log-linear model:.

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Markov Logic networks

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  1. A brief introduction to Markov Logic networks De Raedt, Kersting, Natarajan, Poole: Statistical Relational AI

  2. Markov Networks • Undirected graphical models Smoking Cancer Asthma Cough • Potential functions defined over cliques De Raedt, Kersting, Natarajan, Poole: Statistical Relational AI

  3. Markov Networks • Undirected graphical models Smoking Cancer Asthma Cough • Log-linear model: Weight of Feature i Feature i De Raedt, Kersting, Natarajan, Poole: Statistical Relational AI

  4. Markov Logic: Intuition • 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) De Raedt, Kersting, Natarajan, Poole: Statistical Relational AI

  5. Markov Logic • 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 • An MLN defines a Markov network with • One node for each grounding of each predicatein the MLN • One feature for each grounding of each formula F in the MLN, with the corresponding weight w De Raedt, Kersting, Natarajan, Poole: Statistical Relational AI

  6. Markov Logic Networks • Probability of a world x: • Typed variables and constants greatly reduce size of ground Markov net • Functions, existential quantifiers, etc. • Infinite and continuous domains Weight of formula i No. of true groundings of formula i in x De Raedt, Kersting, Natarajan, Poole: Statistical Relational AI

  7. A possible worlds view Say we have two domain elements Anna and Bob as well as two predicates Friends and Happy De Raedt, Kersting, Natarajan, Poole: Statistical Relational AI

  8. A possible worlds view Logical formulas such as IF Friends(Anna,Bob) THEN Happy(Bob) exclude possible worlds De Raedt, Kersting, Natarajan, Poole: Statistical Relational AI

  9. A possible worlds view Instead of excluding worlds, we want them to become less likely, e.g. De Raedt, Kersting, Natarajan, Poole: Statistical Relational AI

  10. A possible worlds view This is achieved with the following potentials De Raedt, Kersting, Natarajan, Poole: Statistical Relational AI

  11. An possible worlds view Or as log-linear model this is: De Raedt, Kersting, Natarajan, Poole: Statistical Relational AI This can also be viewed as building a graphical model

  12. A graphical example : Friends & Smokers De Raedt, Kersting, Natarajan, Poole: Statistical Relational AI

  13. A graphical example : Friends & Smokers De Raedt, Kersting, Natarajan, Poole: Statistical Relational AI

  14. A graphical example : Friends & Smokers De Raedt, Kersting, Natarajan, Poole: Statistical Relational AI

  15. A graphical example : Friends & Smokers Two constants: Anna (A) and Bob (B) De Raedt, Kersting, Natarajan, Poole: Statistical Relational AI

  16. A graphical example : Friends & Smokers Two constants: Anna (A) and Bob (B) Smokes(A) Smokes(B) Cancer(A) Cancer(B) De Raedt, Kersting, Natarajan, Poole: Statistical Relational AI

  17. A graphical example : Friends & Smokers 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) De Raedt, Kersting, Natarajan, Poole: Statistical Relational AI

  18. A graphical example : Friends & Smokers 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) De Raedt, Kersting, Natarajan, Poole: Statistical Relational AI

  19. A graphical example : Friends & Smokers 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) De Raedt, Kersting, Natarajan, Poole: Statistical Relational AI Alternatively, this can be viewed as a set of weighted ground clauses

  20. A graphical example : Friends & SmokersCan be combined with ideas from graphical models and logic programming to answer queries “efficiently” Two constants: Anna (A) and Bob (B) Friends(A,B) Friends(A,A) Smokes(A) Smokes(B) Friends(B,B) query Cancer(A) Cancer(B) Friends(B,A) P( Cancer(B) | Smokes(A),Friends(A,B),Friends(B,A)) De Raedt, Kersting, Natarajan, Poole: Statistical Relational AI

  21. A graphical example : Friends & SmokersCan be combined with ideas from graphical models and logic programming to answer queries “efficiently” Two constants: Anna (A) and Bob (B) Friends(A,B) Follow the clauses Friends(A,A) Smokes(A) Smokes(B) Friends(B,B) Cancer(A) Cancer(B) Friends(B,A) P( Cancer(B) | Smokes(A),Friends(A,B),Friends(B,A)) De Raedt, Kersting, Natarajan, Poole: Statistical Relational AI

  22. A graphical example : Friends & SmokersCan be combined with ideas from graphical models and logic programming to answer queries “efficiently” Two constants: Anna (A) and Bob (B) Friends(A,B) Follow the clauses Friends(A,A) Smokes(A) Smokes(B) Friends(B,B) Cancer(A) Cancer(B) Friends(B,A) P( Cancer(B) | Smokes(A),Friends(A,B),Friends(B,A)) De Raedt, Kersting, Natarajan, Poole: Statistical Relational AI

  23. A graphical example : Friends & SmokersCan be combined with ideas from graphical models and logic programming to answer queries “efficiently” 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) P( Cancer(B) | Smokes(A),Friends(A,B),Friends(B,A)) De Raedt, Kersting, Natarajan, Poole: Statistical Relational AI

  24. A graphical example : Friends & SmokersCan be combined with ideas from graphical models and logic programming to answer queries “efficiently” 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) P( Cancer(B) | Smokes(A),Friends(A,B),Friends(B,A)) De Raedt, Kersting, Natarajan, Poole: Statistical Relational AI We may not have to consider the complete induced Markov networks

  25. 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.) Relation to Statistical Models De Raedt, Kersting, Natarajan, Poole: Statistical Relational AI

  26. Relation to First-Order Logic • Infinite weights  First-order logic (without function symbolcs) • Satisfiable KB, positive weights Satisfying assignments = Modes of distribution • Markov logic allows contradictions between formulas De Raedt, Kersting, Natarajan, Poole: Statistical Relational AI

  27. Applications Natural language processing, Robotics, Collective Classification, Social Networks, Activity Recognition, … De Raedt, Kersting, Natarajan, Poole: Statistical Relational AI

  28. Information Extraction ParagSingla and Pedro Domingos, “Memory-Efficient Inference in Relational Domains”(AAAI-06). Singla, P., & Domingos, P. (2006). Memory-efficent inference in relatonal domains. In Proceedings of the Twenty-First National Conference on Artificial Intelligence (pp. 500-505). Boston, MA: AAAI Press. H. Poon & P. Domingos, Sound and Efficient Inference with Probabilistic and Deterministic Dependencies”, in Proc. AAAI-06, Boston, MA, 2006. P. Hoifung (2006). Efficent inference. In Proceedings of the Twenty-First National Conference on Artificial Intelligence. De Raedt, Kersting, Natarajan, Poole: Statistical Relational AI

  29. Author Segmentation Title Venue ParagSingla and Pedro Domingos, “Memory-Efficient Inference in Relational Domains”(AAAI-06). Singla, P., & Domingos, P. (2006). Memory-efficent inference in relatonal domains. In Proceedings of the Twenty-First National Conference on Artificial Intelligence (pp. 500-505). Boston, MA: AAAI Press. H. Poon & P. Domingos, Sound and Efficient Inference with Probabilistic and Deterministic Dependencies”, in Proc. AAAI-06, Boston, MA, 2006. P. Hoifung (2006). Efficent inference. In Proceedings of the Twenty-First National Conference on Artificial Intelligence. De Raedt, Kersting, Natarajan, Poole: Statistical Relational AI

  30. Entity Resolution ParagSingla and Pedro Domingos, “Memory-Efficient Inference in Relational Domains”(AAAI-06). Singla, P., & Domingos, P. (2006). Memory-efficent inference in relatonal domains. In Proceedings of the Twenty-First National Conference on Artificial Intelligence (pp. 500-505). Boston, MA: AAAI Press. H. Poon & P. Domingos, Sound and Efficient Inference with Probabilistic and Deterministic Dependencies”, in Proc. AAAI-06, Boston, MA, 2006. P. Hoifung (2006). Efficent inference. In Proceedings of the Twenty-First National Conference on Artificial Intelligence. De Raedt, Kersting, Natarajan, Poole: Statistical Relational AI

  31. Entity Resolution ParagSingla and Pedro Domingos, “Memory-Efficient Inference in Relational Domains”(AAAI-06). Singla, P., & Domingos, P. (2006). Memory-efficent inference in relatonal domains. In Proceedings of the Twenty-First National Conference on Artificial Intelligence (pp. 500-505). Boston, MA: AAAI Press. H. Poon & P. Domingos, Sound and Efficient Inference with Probabilistic and Deterministic Dependencies”, in Proc. AAAI-06, Boston, MA, 2006. P. Hoifung (2006). Efficent inference. In Proceedings of the Twenty-First National Conference on Artificial Intelligence. De Raedt, Kersting, Natarajan, Poole: Statistical Relational AI

  32. Implementing ER using MLNs:Types and Predicates token = {Parag, Singla, and, Pedro, ...} field = {Author, Title, Venue} citation = {C1, C2, ...} position = {0, 1, 2, ...} Token(token, position, citation) InField(position, field, citation) SameField(field, citation, citation) SameCit(citation, citation) De Raedt, Kersting, Natarajan, Poole: Statistical Relational AI

  33. Implementing ER using MLNs:Types and Predicates token = {Parag, Singla, and, Pedro, ...} field = {Author, Title, Venue, ...} citation = {C1, C2, ...} position = {0, 1, 2, ...} Token(token, position, citation) InField(position, field, citation) SameField(field, citation, citation) SameCit(citation, citation) Optional De Raedt, Kersting, Natarajan, Poole: Statistical Relational AI

  34. Implementing ER using MLNs:Types and Predicates token = {Parag, Singla, and, Pedro, ...} field = {Author, Title, Venue} citation = {C1, C2, ...} position = {0, 1, 2, ...} Token(token, position, citation) InField(position, field, citation) SameField(field, citation, citation) SameCit(citation, citation) Will be theevidence De Raedt, Kersting, Natarajan, Poole: Statistical Relational AI

  35. Implementing ER using MLNs:Types and Predicates token = {Parag, Singla, and, Pedro, ...} field = {Author, Title, Venue} citation = {C1, C2, ...} position = {0, 1, 2, ...} Token(token, position, citation) InField(position, field, citation) SameField(field, citation, citation) SameCit(citation, citation) Does the segmentation Do the resolution, i.e., the say which parts are the same Query predicates De Raedt, Kersting, Natarajan, Poole: Statistical Relational AI

  36. Implementing ER using MLNs:Formulas Encoding an HMM/CRF Token(+t,i,c) => InField(i,+f,c) InField(i,+f,c) <=> InField(i+1,+f,c) f != f’ => (!InField(i,+f,c) v !InField(i,+f’,c)) Token(+t,i,c) ^ InField(i,+f,c) ^ Token(+t,i’,c’) ^ InField(i’,+f,c’) => SameField(+f,c,c’) SameField(+f,c,c’) <=> SameCit(c,c’) SameField(f,c,c’) ^ SameField(f,c’,c”) => SameField(f,c,c”) SameCit(c,c’) ^ SameCit(c’,c”) => SameCit(c,c”) De Raedt, Kersting, Natarajan, Poole: Statistical Relational AI

  37. Implementing ER using MLNs:Formulas Encoding an HMM/CRF Token(+t,i,c) => InField(i,+f,c) InField(i,+f,c) <=> InField(i+1,+f,c) f != f’ => (!InField(i,+f,c) v !InField(i,+f’,c)) Token(+t,i,c) ^ InField(i,+f,c) ^ Token(+t,i’,c’) ^ InField(i’,+f,c’) => SameField(+f,c,c’) SameField(+f,c,c’) <=> SameCit(c,c’) SameField(f,c,c’) ^ SameField(f,c’,c”) => SameField(f,c,c”) SameCit(c,c’) ^ SameCit(c’,c”) => SameCit(c,c”) Encoding the ER De Raedt, Kersting, Natarajan, Poole: Statistical Relational AI

  38. Results: Segmentation on Cora De Raedt, Kersting, Natarajan, Poole: Statistical Relational AI

  39. What have we learnt? • Markov logic networks combine first-order logic and Markov networks • Syntax: First-order logic + Weights • Semantics: Templates for Markov networks • There are implementations available • Important real world SRL problems easily formulated as MLNs • Not touched: continuous RVs, tractable versions, compilation, efficient learning, … De Raedt, Kersting, Natarajan, Poole: Statistical Relational AI

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