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Scalable Inference and Learning for High-Level Probabilistic Models Guy Van den Broeck KU Leuven Outline Motivation Why high-level representations? Why high-level reasoning? Intuition: Inference rules Liftability theory:

  1. A Simple Reasoning Problem ... ? Probability that Card52 is Spades 13/51 given that Card1 is QH? [Van den Broeck; AAAI- KRR’15+

  2. Automated Reasoning Let us automate this: 1. Probabilistic graphical model (e.g., factor graph) 2. Probabilistic inference algorithm (e.g., variable elimination or junction tree)

  3. Classical Reasoning A A A B C B C B C D E D E D E F F F Tree Sparse Graph Dense Graph • Higher treewidth • Fewer conditional independencies • Slower inference

  4. Is There Conditional Independence? ... P(Card52 | Card1) ≟ P(Card52 | Card1, Card2)

  5. Is There Conditional Independence? ... ? P(Card52 | Card1) ≟ P(Card52 | Card1, Card2) ? ≟ ?

  6. Is There Conditional Independence? ... ? P(Card52 | Card1) ≟ P(Card52 | Card1, Card2) 13/51 ≟ ?

  7. Is There Conditional Independence? ... ? P(Card52 | Card1) ≟ P(Card52 | Card1, Card2) 13/51 ≟ ?

  8. Is There Conditional Independence? ... ? P(Card52 | Card1) ≟ P(Card52 | Card1, Card2) 13/51 ≠ 12/50

  9. Is There Conditional Independence? ... ? P(Card52 | Card1) ≟ P(Card52 | Card1, Card2) P(Card52 | Card1) ≠ P(Card52 | Card1, Card2) 13/51 ≠ 12/50

  10. Is There Conditional Independence? ... ? P(Card52 | Card1) ≟ P(Card52 | Card1, Card2) P(Card52 | Card1) ≠ P(Card52 | Card1, Card2) 13/51 ≠ 12/50 P(Card52 | Card1, Card2) ≟ P(Card52 | Card1, Card2, Card3)

  11. Is There Conditional Independence? ... ? P(Card52 | Card1) ≟ P(Card52 | Card1, Card2) P(Card52 | Card1) ≠ P(Card52 | Card1, Card2) 13/51 ≠ 12/50 P(Card52 | Card1, Card2) ≟ P(Card52 | Card1, Card2, Card3) P(Card52 | Card1, Card2) ≠ P(Card52 | Card1, Card2, Card3) 12/50 ≠ 12/49

  12. Automated Reasoning Let us automate this: 1. Probabilistic graphical model (e.g., factor graph) is fully connected! (artist's impression) 2. Probabilistic inference algorithm (e.g., variable elimination or junction tree) builds a table with 52 52 rows [Van den Broeck; AAAI- KRR’15+

  13. What's Going On Here? ... ? Probability that Card52 is Spades given that Card1 is QH? [Van den Broeck; AAAI- KRR’15+

  14. What's Going On Here? ... ? Probability that Card52 is Spades 13/51 given that Card1 is QH? [Van den Broeck; AAAI- KRR’15+

  15. What's Going On Here? ... ? Probability that Card52 is Spades given that Card2 is QH? [Van den Broeck; AAAI- KRR’15+

  16. What's Going On Here? ... ? Probability that Card52 is Spades 13/51 given that Card2 is QH? [Van den Broeck; AAAI- KRR’15+

  17. What's Going On Here? ... ? Probability that Card52 is Spades given that Card3 is QH? [Van den Broeck; AAAI- KRR’15+

  18. What's Going On Here? ... ? Probability that Card52 is Spades 13/51 given that Card3 is QH? [Van den Broeck; AAAI- KRR’15+

  19. Tractable Probabilistic Inference ... Which property makes inference tractable? Traditional belief: Independence What's going on here? [Niepert , Van den Broeck; AAAI’14+, *Van den Broeck; AAAI - KRR’15+

  20. Tractable Probabilistic Inference ... Which property makes inference tractable? Traditional belief: Independence What's going on here?  High-level reasoning ⇒ Lifted Inference  Symmetry  Exchangeability [Niepert , Van den Broeck; AAAI’14+, *Van den Broeck; AAAI - KRR’15+

  21. Other Examples of Lifted Inference  Syllogisms & First-order resolution  Reasoning about populations We are investigating a rare disease. The disease is more rare in women, presenting only in one in every two billion women and one in every billion men . Then, assuming there are 3.4 billion men and 3.6 billion women in the world, the probability that more than five people have the disease is [Van den Broeck; AAAI- KRR’15+, *Van den Broeck; PhD‘13+

  22. Equivalent Graphical Model  Statistical relational model (e.g., MLN) 3.14 FacultyPage(x) ∧ Linked(x,y) ⇒ CoursePage(y)  As a probabilistic graphical model:  26 pages; 728 variables; 676 factors  1000 pages; 1,002,000 variables; 1,000,000 factors  Highly intractable? – Lifted inference in milliseconds!

  23. Outline • Motivation – Why high-level representations? – Why high-level reasoning? • Intuition: Inference rules • Liftability theory: Strengths and limitations • Lifting in practice – Approximate symmetries – Lifted learning

  24. Weighted Model Counting • Model = solution to a propositional logic formula Δ • Model counting = #SAT Δ = (Rain ⇒ Cloudy) Rain Cloudy Model? T T Yes T F No F T Yes F F Yes + #SAT = 3

  25. Weighted Model Counting • Model = solution to a propositional logic formula Δ • Model counting = #SAT • Weighted model counting (WMC) – Weights for assignments to variables – Model weight is product of variable weights w(.) Δ = (Rain ⇒ Cloudy) Rain Cloudy Model? Weight T T Yes 1 * 3 = 3 w( R)=1 T F No 0 w(¬R)=2 F T Yes 2 * 3 = 6 w( C)=3 w(¬C)=5 F F Yes 2 * 5 = 10 + #SAT = 3

  26. Weighted Model Counting • Model = solution to a propositional logic formula Δ • Model counting = #SAT • Weighted model counting (WMC) – Weights for assignments to variables – Model weight is product of variable weights w(.) Δ = (Rain ⇒ Cloudy) Rain Cloudy Model? Weight T T Yes 1 * 3 = 3 w( R)=1 T F No 0 w(¬R)=2 F T Yes 2 * 3 = 6 w( C)=3 w(¬C)=5 F F Yes 2 * 5 = 10 + + #SAT = 3 WMC = 19

  27. Assembly language for probabilistic reasoning Factor graphs Probabilistic Bayesian networks logic programs Relational Bayesian Probabilistic Markov Logic networks databases Weighted Model Counting

  28. Weighted First-Order Model Counting Model = solution to first-order logic formula Δ Δ = ∀ d (Rain(d) ⇒ Cloudy(d)) Days = {Monday}

  29. Weighted First-Order Model Counting Model = solution to first-order logic formula Δ Δ = ∀ d (Rain(d) Rain(M) Cloudy(M) Model? ⇒ Cloudy(d)) T T Yes T F No Days = {Monday} F T Yes F F Yes + #SAT = 3

  30. Weighted First-Order Model Counting Model = solution to first-order logic formula Δ Δ = ∀ d (Rain(d) Rain(M) Cloudy(M) Rain(T) Cloudy(T) Model? ⇒ Cloudy(d)) T T T T Yes T F T T No F T T T Yes Days = {Monday F F T T Yes Tuesday } T T T F No T F T F No F T T F No F F T F No T T F T Yes T F F T No F T F T Yes F F F T Yes T T F F Yes T F F F No F T F F Yes F F F F Yes

  31. Weighted First-Order Model Counting Model = solution to first-order logic formula Δ Δ = ∀ d (Rain(d) Rain(M) Cloudy(M) Rain(T) Cloudy(T) Model? ⇒ Cloudy(d)) T T T T Yes T F T T No F T T T Yes Days = {Monday F F T T Yes Tuesday } T T T F No T F T F No F T T F No F F T F No T T F T Yes T F F T No F T F T Yes F F F T Yes T T F F Yes T F F F No F T F F Yes F F F F Yes + #SAT = 9

  32. Weighted First-Order Model Counting Model = solution to first-order logic formula Δ Δ = ∀ d (Rain(d) Rain(M) Cloudy(M) Rain(T) Cloudy(T) Model? Weight ⇒ Cloudy(d)) T T T T Yes 1 * 1 * 3 * 3 = 9 T F T T No 0 F T T T Yes 2 * 1* 3 * 3 = 18 Days = {Monday F F T T Yes 2 * 1 * 5 * 3 = 30 Tuesday } T T T F No 0 T F T F No 0 w( R)=1 F T T F No 0 w(¬R)=2 F F T F No 0 w( C)=3 T T F T Yes 1 * 2 * 3 * 3 = 18 w(¬C)=5 T F F T No 0 F T F T Yes 2 * 2 * 3 * 3 = 36 F F F T Yes 2 * 2 * 5 * 3 = 60 T T F F Yes 1 * 2 * 3 * 5 = 30 T F F F No 0 F T F F Yes 2 * 2 * 3 * 5 = 60 F F F F Yes 2 * 2 * 5 * 5 = 100 + #SAT = 9

  33. Weighted First-Order Model Counting Model = solution to first-order logic formula Δ Δ = ∀ d (Rain(d) Rain(M) Cloudy(M) Rain(T) Cloudy(T) Model? Weight ⇒ Cloudy(d)) T T T T Yes 1 * 1 * 3 * 3 = 9 T F T T No 0 F T T T Yes 2 * 1* 3 * 3 = 18 Days = {Monday F F T T Yes 2 * 1 * 5 * 3 = 30 Tuesday } T T T F No 0 T F T F No 0 w( R)=1 F T T F No 0 w(¬R)=2 F F T F No 0 w( C)=3 T T F T Yes 1 * 2 * 3 * 3 = 18 w(¬C)=5 T F F T No 0 F T F T Yes 2 * 2 * 3 * 3 = 36 F F F T Yes 2 * 2 * 5 * 3 = 60 T T F F Yes 1 * 2 * 3 * 5 = 30 T F F F No 0 F T F F Yes 2 * 2 * 3 * 5 = 60 F F F F Yes 2 * 2 * 5 * 5 = 100 + + WFOMC = 361 #SAT = 9

  34. Assembly language for high-level probabilistic reasoning Probabilistic Parfactor graphs logic programs Relational Bayesian Probabilistic Markov Logic networks databases Weighted First-Order Model Counting [VdB et al.; IJCAI’11, PhD’13, KR’14, UAI’14]

  35. WFOMC Inference: Example • FO-Model Counting: w(R) = w(¬R) = 1 • Apply inference rules backwards (step 4-3-2-1)

  36. WFOMC Inference: Example • FO-Model Counting: w(R) = w(¬R) = 1 • Apply inference rules backwards (step 4-3-2-1) Δ = (Stress(Alice) ⇒ Smokes(Alice)) 4. Domain = {Alice}

  37. WFOMC Inference: Example • FO-Model Counting: w(R) = w(¬R) = 1 • Apply inference rules backwards (step 4-3-2-1) Δ = (Stress(Alice) ⇒ Smokes(Alice)) 4. Domain = {Alice} → 3 models

  38. WFOMC Inference: Example • FO-Model Counting: w(R) = w(¬R) = 1 • Apply inference rules backwards (step 4-3-2-1) Δ = (Stress(Alice) ⇒ Smokes(Alice)) 4. Domain = {Alice} → 3 models Δ = ∀ x, (Stress(x) ⇒ Smokes(x)) 3. Domain = {n people}

  39. WFOMC Inference: Example • FO-Model Counting: w(R) = w(¬R) = 1 • Apply inference rules backwards (step 4-3-2-1) Δ = (Stress(Alice) ⇒ Smokes(Alice)) 4. Domain = {Alice} → 3 models Δ = ∀ x, (Stress(x) ⇒ Smokes(x)) 3. Domain = {n people} → 3 n models

  40. WFOMC Inference: Example Δ = ∀ x, (Stress(x) ⇒ Smokes(x)) 3. Domain = {n people} → 3 n models

  41. WFOMC Inference: Example Δ = ∀ x, (Stress(x) ⇒ Smokes(x)) 3. Domain = {n people} → 3 n models Δ = ∀ y, (ParentOf(y) ∧ Female ⇒ MotherOf(y)) 2. D = {n people}

  42. WFOMC Inference: Example Δ = ∀ x, (Stress(x) ⇒ Smokes(x)) 3. Domain = {n people} → 3 n models Δ = ∀ y, (ParentOf(y) ∧ Female ⇒ MotherOf(y)) 2. D = {n people} → 3 n models Δ = ∀ y, (ParentOf(y) ⇒ MotherOf(y)) If Female = true?

  43. WFOMC Inference: Example Δ = ∀ x, (Stress(x) ⇒ Smokes(x)) 3. Domain = {n people} → 3 n models Δ = ∀ y, (ParentOf(y) ∧ Female ⇒ MotherOf(y)) 2. D = {n people} → 3 n models Δ = ∀ y, (ParentOf(y) ⇒ MotherOf(y)) If Female = true? → 4 n models If Female = false? Δ = true

  44. WFOMC Inference: Example Δ = ∀ x, (Stress(x) ⇒ Smokes(x)) 3. Domain = {n people} → 3 n models Δ = ∀ y, (ParentOf(y) ∧ Female ⇒ MotherOf(y)) 2. D = {n people} → 3 n models Δ = ∀ y, (ParentOf(y) ⇒ MotherOf(y)) If Female = true? → 4 n models If Female = false? Δ = true → 3 n + 4 n models

  45. WFOMC Inference: Example Δ = ∀ x, (Stress(x) ⇒ Smokes(x)) 3. Domain = {n people} → 3 n models Δ = ∀ y, (ParentOf(y) ∧ Female ⇒ MotherOf(y)) 2. D = {n people} → 3 n models Δ = ∀ y, (ParentOf(y) ⇒ MotherOf(y)) If Female = true? → 4 n models If Female = false? Δ = true → 3 n + 4 n models Δ = ∀ x,y, (ParentOf(x,y) ∧ Female(x) ⇒ MotherOf(x,y)) 1. D = {n people}

  46. WFOMC Inference: Example Δ = ∀ x, (Stress(x) ⇒ Smokes(x)) 3. Domain = {n people} → 3 n models Δ = ∀ y, (ParentOf(y) ∧ Female ⇒ MotherOf(y)) 2. D = {n people} → 3 n models Δ = ∀ y, (ParentOf(y) ⇒ MotherOf(y)) If Female = true? → 4 n models If Female = false? Δ = true → 3 n + 4 n models Δ = ∀ x,y, (ParentOf(x,y) ∧ Female(x) ⇒ MotherOf(x,y)) 1. D = {n people} n models → (3 n + 4 n )

  47. Atom Counting: Example Δ = ∀ x,y, (Smokes(x) ∧ Friends(x,y) ⇒ Smokes(y)) Domain = {n people}

  48. Atom Counting: Example Δ = ∀ x,y, (Smokes(x) ∧ Friends(x,y) ⇒ Smokes(y)) Domain = {n people}  If we know precisely who smokes, and there are k smokers? Database: Smokes Friends Smokes Smokes(Alice) = 1 Smokes(Bob) = 0 Smokes(Charlie) = 0 k k Smokes(Dave) = 1 Smokes(Eve) = 0 ... n-k n-k

  49. Atom Counting: Example Δ = ∀ x,y, (Smokes(x) ∧ Friends(x,y) ⇒ Smokes(y)) Domain = {n people}  If we know precisely who smokes, and there are k smokers? Database: Smokes Friends Smokes Smokes(Alice) = 1 Smokes(Bob) = 0 Smokes(Charlie) = 0 k k Smokes(Dave) = 1 Smokes(Eve) = 0 ... n-k n-k

  50. Atom Counting: Example Δ = ∀ x,y, (Smokes(x) ∧ Friends(x,y) ⇒ Smokes(y)) Domain = {n people}  If we know precisely who smokes, and there are k smokers? Database: Smokes Friends Smokes Smokes(Alice) = 1 Smokes(Bob) = 0 Smokes(Charlie) = 0 k k Smokes(Dave) = 1 Smokes(Eve) = 0 ... n-k n-k

  51. Atom Counting: Example Δ = ∀ x,y, (Smokes(x) ∧ Friends(x,y) ⇒ Smokes(y)) Domain = {n people}  If we know precisely who smokes, and there are k smokers? Database: Smokes Friends Smokes Smokes(Alice) = 1 Smokes(Bob) = 0 Smokes(Charlie) = 0 k k Smokes(Dave) = 1 Smokes(Eve) = 0 ... n-k n-k

  52. Atom Counting: Example Δ = ∀ x,y, (Smokes(x) ∧ Friends(x,y) ⇒ Smokes(y)) Domain = {n people}  If we know precisely who smokes, and there are k smokers? Database: Smokes Friends Smokes Smokes(Alice) = 1 Smokes(Bob) = 0 Smokes(Charlie) = 0 k k Smokes(Dave) = 1 Smokes(Eve) = 0 ... n-k n-k

  53. Atom Counting: Example Δ = ∀ x,y, (Smokes(x) ∧ Friends(x,y) ⇒ Smokes(y)) Domain = {n people}  If we know precisely who smokes, and there are k smokers? Database: Smokes Friends Smokes Smokes(Alice) = 1 Smokes(Bob) = 0 Smokes(Charlie) = 0 k k Smokes(Dave) = 1 Smokes(Eve) = 0 ... n-k n-k

  54. Atom Counting: Example Δ = ∀ x,y, (Smokes(x) ∧ Friends(x,y) ⇒ Smokes(y)) Domain = {n people}  If we know precisely who smokes, and there are k smokers? Database: Smokes Friends Smokes Smokes(Alice) = 1 Smokes(Bob) = 0 Smokes(Charlie) = 0 k k Smokes(Dave) = 1 Smokes(Eve) = 0 ... n-k n-k

  55. Atom Counting: Example Δ = ∀ x,y, (Smokes(x) ∧ Friends(x,y) ⇒ Smokes(y)) Domain = {n people}  If we know precisely who smokes, and there are k smokers? Database: Smokes Friends Smokes Smokes(Alice) = 1 Smokes(Bob) = 0 Smokes(Charlie) = 0 k k Smokes(Dave) = 1 Smokes(Eve) = 0 ... n-k n-k

  56. Atom Counting: Example Δ = ∀ x,y, (Smokes(x) ∧ Friends(x,y) ⇒ Smokes(y)) Domain = {n people}  If we know precisely who smokes, and there are k smokers? Database: Smokes Friends Smokes Smokes(Alice) = 1 Smokes(Bob) = 0 Smokes(Charlie) = 0 k k Smokes(Dave) = 1 Smokes(Eve) = 0 ... n-k n-k

  57. Atom Counting: Example Δ = ∀ x,y, (Smokes(x) ∧ Friends(x,y) ⇒ Smokes(y)) Domain = {n people}  If we know precisely who smokes, and there are k smokers? Database: Smokes Friends Smokes Smokes(Alice) = 1 Smokes(Bob) = 0 Smokes(Charlie) = 0 k k Smokes(Dave) = 1 Smokes(Eve) = 0 ... n-k n-k → models

  58. Atom Counting: Example Δ = ∀ x,y, (Smokes(x) ∧ Friends(x,y) ⇒ Smokes(y)) Domain = {n people}  If we know precisely who smokes, and there are k smokers? Database: Smokes Friends Smokes Smokes(Alice) = 1 Smokes(Bob) = 0 Smokes(Charlie) = 0 k k Smokes(Dave) = 1 Smokes(Eve) = 0 ... n-k n-k → models  If we know that there are k smokers?

  59. Atom Counting: Example Δ = ∀ x,y, (Smokes(x) ∧ Friends(x,y) ⇒ Smokes(y)) Domain = {n people}  If we know precisely who smokes, and there are k smokers? Database: Smokes Friends Smokes Smokes(Alice) = 1 Smokes(Bob) = 0 Smokes(Charlie) = 0 k k Smokes(Dave) = 1 Smokes(Eve) = 0 ... n-k n-k → models  If we know that there are k smokers? → models

  60. Atom Counting: Example Δ = ∀ x,y, (Smokes(x) ∧ Friends(x,y) ⇒ Smokes(y)) Domain = {n people}  If we know precisely who smokes, and there are k smokers? Database: Smokes Friends Smokes Smokes(Alice) = 1 Smokes(Bob) = 0 Smokes(Charlie) = 0 k k Smokes(Dave) = 1 Smokes(Eve) = 0 ... n-k n-k → models  If we know that there are k smokers? → models  In total…

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AIM Form Training May-Aug 2020 Form Version Medically Ready ForceReady Medical Force

AIM Form Training May-Aug 2020 Form Version Medically Ready ForceReady Medical Force

TSWF Pulmonary CPG AIM Form Training May-Aug 2020 Form Version Medically Ready ForceReady Medical Force Disclaimer Content shown is from an AHLTA Training System (ATS) and does not contain actual patient data. Medically Ready

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