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Lifted Probabilistic Inference by First-Order Knowledge Compilation Guy Van den Broeck Nima Taghipour Wannes Meert Jesse Davis Luc De Raedt Lifted Inference in Probabilistic Logical Models - Tutorial - IJCAI11 18/07/11 Outline Overview

  1. Lifted Probabilistic Inference by First-Order Knowledge Compilation Guy Van den Broeck Nima Taghipour Wannes Meert Jesse Davis Luc De Raedt Lifted Inference in Probabilistic Logical Models - Tutorial - IJCAI11 18/07/11

  2. Outline ● Overview Approach ● First-Order d-DNNF Circuits ● First-Order Knowledge Compilation ● Experiments ● Conclusions 18/07/11 First-Order Knowledge Compilation 2

  3. Outline ● Overview Approach ● First-Order d-DNNF Circuits ● First-Order Knowledge Compilation ● Experiments ● Conclusions 18/07/11 First-Order Knowledge Compilation 3

  4. Context Variable Belief Knowledge Elimination Propagation Compilation Ground [Zhang94] [Pearl82] [Darwiche03] Lifted [Poole03] [Singla08] Our approach 18/07/11 First-Order Knowledge Compilation 4

  5. Advantages of Knowledge Compilation ● Compile once, then run polytime inference for multiple queries and evidence ● Efficient data structures ● Principled logical approach ● Exploits context-specific independences ● State of the art for exact inference in ● Bayesian networks ● Statistical relational learning ● Used in many domains, not just probabilistic reasoning 18/07/11 First-Order Knowledge Compilation 5

  6. Question? ● Can we lift knowledge compilation to a first-order setting? ● First step taken: first-order d-DNNFs for ● weighted first-order model counting ● lifted probabilistic inference ● Many open questions remaining! 18/07/11 First-Order Knowledge Compilation 6

  7. What is Lifted Inference? 18/07/11 First-Order Knowledge Compilation 7

  8. What is Lifted Inference? ● Variables X,Y range over domain People ● Represents propositional model for given domain (50 people) ● Propositional inference in factor graph is expensive ● However: symmetries 18/07/11 First-Order Knowledge Compilation 8

  9. What is Lifted Inference? ● We compile to a circuit independent of |People| ● Inference linear in |People| → Lifted Inference 18/07/11 First-Order Knowledge Compilation 9

  10. Knowledge Compilation Bayesian Network Factor Graph MLN ... 18/07/11 First-Order Knowledge Compilation 10

  11. Knowledge Compilation Bayesian Network 1 Factor Graph Weighted CNF MLN ... ● Step : Convert model to weighted CNF 1 18/07/11 First-Order Knowledge Compilation 11

  12. Knowledge Compilation Bayesian Network 1 2 Factor Graph Weighted d-DNNF CNF Circuit MLN ... ● Step : Convert model to weighted CNF 1 ● Step : Convert CNF to d-DNNF circuit 2 18/07/11 First-Order Knowledge Compilation 12

  13. Knowledge Compilation Bayesian Network 1 2 3 Factor Graph Weighted d-DNNF Weighted CNF Circuit Model Counting MLN ... ● Step : Convert model to weighted CNF 1 ● Step : Convert CNF to d-DNNF circuit 2 ● Step : Perform weighted model counting 3 18/07/11 First-Order Knowledge Compilation 13

  14. Our Approach: First-Order Knowledge Compilation: Parfactor 1 2 3 Graph Weighted FO d-DNNF Weighted FO MLN FO CNF Circuit Model Counting ... ● Step : Convert model to weighted FO CNF 1 ● Step : Convert CNF to FO d-DNNF circuit 2 ● Step : Perform weighted FO model counting 3 18/07/11 First-Order Knowledge Compilation 14

  15. Step 1: Converting to Weighted FO CNF MLN 18/07/11 First-Order Knowledge Compilation 15

  16. Step 1: Converting to Weighted FO CNF MLN Weighted FO Theory 18/07/11 First-Order Knowledge Compilation 16

  17. Step 1: Converting to Weighted FO CNF MLN Weighted FO Theory Weighted FO CNF 18/07/11 First-Order Knowledge Compilation 17

  18. Step 3: Weighted FO Model Counting ● Weight function on ground atoms 18/07/11 First-Order Knowledge Compilation 18

  19. Step 3: Weighted FO Model Counting ● Weight function on ground atoms ● Weight of a model (possible world) 18/07/11 First-Order Knowledge Compilation 19

  20. Step 3: Weighted FO Model Counting ● Weight function on ground atoms ● Weight of a model (possible world) ● Weight of all models is 18/07/11 First-Order Knowledge Compilation 20

  21. Step 3: Weighted FO Model Counting ● Weight function on ground atoms ● Weight of a model (possible world) ● Weight of all models is Weight of models where Alice smokes is 18/07/11 First-Order Knowledge Compilation 21

  22. Step 3: Weighted FO Model Counting ● Weight function on ground atoms ● Weight of a model (possible world) ● Weight of all models is Weight of models where Alice smokes is ● 18/07/11 First-Order Knowledge Compilation 22

  23. Outline ● Overview Approach ● First-Order d-DNNF Circuits ● First-Order Knowledge Compilation ● Experiments ● Conclusions 18/07/11 First-Order Knowledge Compilation 23

  24. Propositional d-DNNF Circuits [Darwiche and Marquis, 2002] Logical theory: 18/07/11 First-Order Knowledge Compilation 24

  25. Propositional d-DNNF Circuits [Darwiche and Marquis, 2002] Logical theory: Literal (leaf) 18/07/11 First-Order Knowledge Compilation 25

  26. Propositional d-DNNF Circuits [Darwiche and Marquis, 2002] Logical theory: Logical operators (inner node) Literal (leaf) 18/07/11 First-Order Knowledge Compilation 26

  27. Propositional d-DNNF Circuits [Darwiche and Marquis, 2002] Logical theory: Logical operators (inner node) Deterministic disjunction Literal (leaf) 18/07/11 First-Order Knowledge Compilation 27

  28. Propositional d-DNNF Circuits [Darwiche and Marquis, 2002] Logical theory: Logical operators (inner node) Deterministic disjunction Decomposable conjunction Literal (leaf) 18/07/11 First-Order Knowledge Compilation 28

  29. First-Order d-DNNF Circuits Logical Theory: 18/07/11 First-Order Knowledge Compilation 29

  30. First-Order d-DNNF Circuits Logical Theory: First-Order Literal (leaf) 18/07/11 First-Order Knowledge Compilation 30

  31. First-Order d-DNNF Circuits Logical Theory: First-Order Literal (leaf) ● Deterministic disjunction ● Decomposable conjunction ● 3 additional first-order operators (inner nodes) 18/07/11 First-Order Knowledge Compilation 31

  32. Outline ● Overview Approach ● First-Order d-DNNF Circuits ● First-Order Knowledge Compilation ● Experiments ● Conclusions 18/07/11 First-Order Knowledge Compilation 32

  33. Step 2: Our Compilation Algorithm ● Recursively apply Weighted FO d-DNNF FO CNF Circuit ● Unit Propagation ● Independence ● Inclusion-Exclusion (Shannon Decomposition) ● Shattering ● Independent Partial Grounding ● Atom Counting ● (Grounding) 18/07/11 First-Order Knowledge Compilation 33

  34. Step 2: Our Compilation Algorithm ● Recursively apply Weighted FO d-DNNF FO CNF Circuit ● Unit Propagation ● Independence ● Inclusion-Exclusion (Shannon Decomposition) ● Shattering ● Independent Partial Grounding ● Atom Counting ● (Grounding) Generate first-order operators in inner nodes 18/07/11 First-Order Knowledge Compilation 34

  35. Step 2: Our Compilation Algorithm ● Recursively apply Weighted FO d-DNNF FO CNF Circuit ● Unit Propagation ● Independence ● Inclusion-Exclusion (Shannon Decomposition) ● Shattering ● Independent Partial Grounding ● Atom Counting ● (Grounding) 18/07/11 First-Order Knowledge Compilation 35

  36. Unit Propagation 18/07/11 First-Order Knowledge Compilation 36

  37. Unit Propagation Unit clause 18/07/11 First-Order Knowledge Compilation 37

  38. Unit Propagation Unit clause 18/07/11 First-Order Knowledge Compilation 38

  39. Unit Propagation Clauses split w.r.t. unit clause atom 'residuals' → independent 18/07/11 First-Order Knowledge Compilation 39

  40. Unit Propagation Resolvent of unit and 2 nd clause 18/07/11 First-Order Knowledge Compilation 40

  41. Step 2: Our Compilation Algorithm ● Recursively apply Weighted FO d-DNNF FO CNF Circuit ● Unit Propagation ● Independence ● Inclusion-Exclusion (Shannon Decomposition) ● Shattering ● Independent Partial Grounding ● Atom Counting ● (Grounding) 18/07/11 First-Order Knowledge Compilation 41

  42. Atom Counting 18/07/11 First-Order Knowledge Compilation 42

  43. Atom Counting Atom with one logical variable X ∈ {luc,jesse} 18/07/11 First-Order Knowledge Compilation 43

  44. Atom Counting Atom with 1 logical variable 18/07/11 First-Order Knowledge Compilation 44

  45. Atom Counting All partial interpretations for fun(X) - deterministic - 2 |People| 18/07/11 First-Order Knowledge Compilation 45

  46. Atom Counting Same weighted model count 18/07/11 First-Order Knowledge Compilation 46

  47. Atom Counting 2 |People| → |People|+1 18/07/11 First-Order Knowledge Compilation 47

  48. Atom Counting Isomorphic circuits 18/07/11 First-Order Knowledge Compilation 48

  49. Atom Counting |People|+1 → 1 18/07/11 First-Order Knowledge Compilation 49

  50. Atom Counting 18/07/11 First-Order Knowledge Compilation 50

  51. Outline ● Overview Approach ● First-Order d-DNNF Circuits ● First-Order Knowledge Compilation ● Experiments ● Conclusions 18/07/11 First-Order Knowledge Compilation 51

  52. Evaluated Models ● Sick Death [de Salvo Braz 2005] ● WebKB [Lowd 2007] ● Competing Workshops [Milch 2008] ● Workshop Attributes [Milch 2008] ● Friends Smoker [Singla 2008] ● Friends Smoker Drinker 18/07/11 First-Order Knowledge Compilation 52

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