1. 1 MPE and Partial Inversion in Lifted Probabilistic Variable Elimination Rodrigo de Salvo Braz University of Illinois at Urbana-Champaign with Eyal Amir and Dan Roth

2. Page 2 Repetitive patterns in graphical models sick(mary,measles) hospital(mary) epidemic(measles)epidemic(flu) sick(mary,flu) … … sick(bob,measles) hospital(bob) sick(bob,flu) …… … …… ……

3. Page 3 Repetitive patterns in graphical models sick(mary,measles) hospital(mary) epidemic(measles)epidemic(flu) sick(mary,flu) … … sick(bob,measles) hospital(bob) sick(bob,flu) …… … …… ……   sick(mary,measles), epidemic(measles))

4. Page 4 Repetitive patterns in graphical models sick(mary,measles) hospital(mary) epidemic(measles)epidemic(flu) sick(mary,flu) … … sick(bob,measles) hospital(bob) sick(bob,flu) …… … …… ……

5. Page 5 Lots of Redundancy! sick(mary,measles) hospital(mary) epidemic(measles)epidemic(flu) sick(mary,flu) … … sick(bob,measles) hospital(bob) sick(bob,flu) …… … …… ……

6. Page 6 Representing structure sick(mary,measles) epidemic(measles)epidemic(flu) sick(mary,flu) … … sick(bob,measles)sick(bob,flu) …… …… sick(P,D) epidemic(D) Poole (2003) named these parfactors, for “parameterized factors”

7. Page 7 Parfactor sick(Person,Disease) epidemic(Disease) 8 Person, Disease   sick(Person,Disease), epidemic(Disease))

8. Page 8 Parfactor sick(Person,Disease) epidemic(Disease) 8 Person, Disease   sick(Person,Disease), epidemic(Disease)), Person  mary, Disease  flu

9. Page 9 Lifted Probabilistic Inference Goal: to perform inference at the first-order level, without resorting to grounding. First-Order Variable Elimination (FOVE): a generalization of Variable Elimination in propositional graphical models. Eliminates classes of random variables at once.

10. Page 10 Inference - Inversion Elimination (IE) P(hospital(mary) | sick(mary, measles)) = ? hospital(mary) sick(mary, D) epidemic(D)

11. Page 11 Inference - Inversion Elimination (IE) P(hospital(mary) | sick(mary, measles)) = ? hospital(mary) sick(mary, D) epidemic(D) = Unification

12. Page 12 Inference - Inversion Elimination (IE) P(hospital(mary) | sick(mary, measles)) = ? sick(mary,measles) hospital(mary) sick(mary, D) D  measles epidemic(measles)epidemic(D) D  measles

13. Page 13 Inference - Inversion Elimination (IE) P(hospital(mary) | sick(mary, measles)) = ? sick(mary,measles) hospital(mary) sick(mary, D) D  measles epidemic(measles)epidemic(D) D  measles =

14. Page 14 Inference - Inversion Elimination (IE) P(hospital(mary) | sick(mary, measles)) = ? sick(mary,measles) hospital(mary) sick(mary, D) D  measles epidemic(measles)epidemic(D) D  measles

15. Page 15 Inference - Inversion Elimination (IE) P(hospital(mary) | sick(mary, measles)) = ? sick(mary,measles) hospital(mary) sick(mary, D) D  measles epidemic(D) D  measles

16. Page 16 Inference - Inversion Elimination (IE) hospital(mary) sick(mary, D) D  measles epidemic(D) D  measles P(hospital(mary) | sick(mary, measles)) = ?

17. Page 17 Inference - Inversion Elimination (IE) P(hospital(mary) | sick(mary, measles)) = ? hospital(mary) sick(mary, D) D  measles

18. Page 18 Inference - Inversion Elimination (IE) P(hospital(mary) | sick(mary, measles)) = ? hospital(mary)

19. Page 19 Inversion Elimination Joint       (A   ) Example  X   (p(X))  X,Y   (p(X),q(X,Y)) Marginalization by eliminating class q(X,Y):  q(X,Y)  X   (p(X))  X,Y   (p(X),q(X,Y))  X   (p(X))  q(X,Y)  X,Y   (p(X),q(X,Y))

20. Page 20 Inversion Elimination  q(X,Y)  X,Y   (p(X),q(X,Y)) =  X,Y  q(X,Y)   (p(X),q(X,Y)) =  X,Y   (p(X)) =  X    Y  (p(X)) =  X   (p(X)) * depends on certain conditions *

21. Page 21 Inversion Elimination - Conditions - I Eliminated atom must contain all logical variables in parfactors involved. sick(P,D) epidemic(D)

22. Page 22 Inversion Elimination - Conditions - I Eliminated atom must contain all logical variables in parfactors involved. sick(P,D) epidemic(D) Ok, contains both P and D

23. Page 23 Inversion Elimination - Conditions - I Eliminated atom must contain all logical variables in parfactors involved. sick(P,D) epidemic(D) Not Ok, missing P sick(P,D)

24. Page 24 Inversion Elimination - Conditions - I Eliminated atom must contain all logical variables in parfactors involved. q(Y,Z) p(X,Y) No atom can be eliminated

25. Page 25 Inversion Elimination - Conditions - I … sick(mary, flu) epidemic(flu) sick(mary, rubella) epidemic(rubella) … sick(mary, D) epidemic(D) D  measles Eliminated atom must contain all logical variables - guarantees that subproblems are disjoint.

26. Page 26 Inversion Elimination - Conditions - II epidemic(measles) epidemic(flu) epidemic(D2) epidemic(D1) epidemic(rubella) … Inversion Elimination Not Ok D1  D2 Requires eliminated RVs to occur in separate instances of parfactor

27. Page 27  e(D)  D1  D2  (e(D 1 ),e(D 2 )) =  e(D)  (0,0) #(0,0) in e(D),D1  D2  (0,1) #(0,1) in e(D),D1  D2  (1,0) #(1,0) in e(D),D1  D2  (1,1) #(1,1) in e(D),D1  D2 =  e(D)  v  (v) #v in e(D),D1  D2 Counting Elimination - A Combinatorial Approach =  ( )  v  (v) #v in e(D),D1  D2 (from i) |e(D)| i i=0

28. Page 28 No shared logical variables between atoms, so counting can be done independently  (epidemic(D 1, Region), epidemic(D 2, Region)) Counting Elimination - A Combinatorial Approach

29. Page 29 Uncovered by Inversion and Counting Eliminating epidemic from   epidemic(Disease1,Region), epidemic(Disease2,Region), donations) No logical variable in all atoms, so no Inversion Elimination Shared logical variables, so no Counting Elimination

30. Page 30 Partial Inversion  e(D,R)  D1  D2,R   e(D1,R), e(D2,R), d )  e(D,R)  D1  D2,R   e(D1,R), e(D2,R), d )  R  e(D,r)  D1  D2   e(D1,r), e(D2,r), d )  R  ’  d ) =  ’  d ) |R| =  ’’  d ) Inversion elimination is the case where all logical variables are inverted and subproblem is propositional.

31. Page 31 Partial Inversion, graphically epidemic(D2,r 1 ) epidemic(D1,r 1 ) D1  D2 donations epidemic(D2,R) epidemic(D1,R) D1  D2 donations epidemic(D2,r 10 ) epidemic(D1,r 10 ) D1  D2 … … Each instance a counting elimination problem

32. Page 32 Partial inversion conditions Conditioned subsets must be disjoint   friends(P1, P2), friends(P2,P1), smoke(P1), smoke(P2) ) Doesn’t work because subproblems share instances of friends.

33. Page 33 Second contribution: Lifted MPE In propositional case, MPE done by factors containing MPE of eliminated variables. AB C D

34. Page 34 MPE AB D BD  000.3C=1 010.2C=1 100.5C=0 110.9C=1 In propositional case, MPE done by factors containing MPE of eliminated variables.

35. Page 35 MPE AB B  00.5C=1,D=0 11.4C=1,D=1 In propositional case, MPE done by factors containing MPE of eliminated variables.

36. Page 36 MPE A A  MPE(B,C,D) 00.9B=0,C=1,D=0 10.7B=1,C=1,D=1 In propositional case, MPE done by factors containing MPE of eliminated variables.

37. Page 37 MPE  0.9A=0,B=1,C=1,D=1 In propositional case, MPE done by factors containing MPE of eliminated variables.

38. Page 38 MPE Same idea in First-order case But factors are quantified and so are assignments: p(X)q(X,Y)  MPE 000.3 r(X,Y) = 1 010.2 r(X,Y) = 1 100.5 r(X,Y) = 0 110.9 r(X,Y) = 1 8 X, Y   p(X), q(X,Y))

39. Page 39 MPE After Inversion Elimination of q(X,Y): p(X)q(X,Y)  MPE 000.3 r(X,Y) = 1 010.9 r(X,Y) = 1 100.5 r(X,Y) = 0 110.3 r(X,Y) = 1 8 X, Y   p(X), q(X,Y)) p(X) ’’ MPE 00.05 8 Y q(X,Y) = 1, r(X,Y) = 1 10.02 8 Y q(X,Y) = 0, r(X,Y) = 1 8 X  ’  p(X)) Lifted assignments

40. Page 40 MPE After Inversion Elimination of p(X): 8 X  ’  p(X))  ’’ MPE 0.009 8 X 8 Y p(X) = 0, q(X,Y) = 1, r(X,Y) = 0  ’’  ) p(X) ’’ MPE 00.05 8 Y q(X,Y) = 1, r(X,Y) = 1 10.02 8 Y q(X,Y) = 0, r(X,Y) = 1

41. Page 41 MPE After Counting Elimination of e: e(D1)e(D2)  MPE 000.3 r(D1,D2) = 1 010.9 r(D1,D2) = 1 100.5 r(D1,D2) = 0 110.3 r(D1,D2) = 1 8 D1, D2   e(D1), e(D2)) ’’ MPE 0.05 9 38 (D1=0,D2=0) e(D1)=0, e(D2) = 1, r(D1,D2) = 1 9 12 (D1=0,D2=1) e(D1)=1, e(D2) = 1, r(D1,D2) = 1 9 15 (D1=1,D2=0) e(D1)=1, e(D2) = 0, r(D1,D2) = 0 9 25 (D1=1,D2=1) e(D1)=0, e(D2) = 0, r(D1,D2) = 1 ’)’)

42. Page 42 Conclusions Partial Inversion: More general algorithm, subsumes Inversion elimination Lifted MPE same idea as in propositional VE, but with Lifted assignments: describe sets of basic assignments Universally quantified comes from Inversion Existentially quantified comes from Counting elimination Ultimate goal: To perform lifted probabilistic inference in way similar to logic inference: without grounding and at a higher level.

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