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 This talk Lifted Probabilistic Inference: performing probabilistic inference from a first-order level Two contributions: Partial inversion: more general algorithm compared to previous work (IJCAI '05) MPE and Lifted assignments

3. Page 3 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” Atom Logical variable

4. Page 4 Parfactor sick(Person,Disease) epidemic(Disease) 8 Person, Disease   sick(Person,Disease), epidemic(Disease))

5. Page 5 Parfactor sick(Person,Disease) epidemic(Disease) 8 Person, Disease   sick(Person,Disease), epidemic(Disease)), Person  mary, Disease  flu

6. Page 6 Lifted Probabilistic Inference First-Order Variable Elimination (FOVE): a generalization of Variable Elimination in propositional graphical models. Eliminates classes of random variables at once.

7. Page 7 FOVE P(hospital(mary) | sick(mary, measles)) = ? sick(mary,measles) hospital(mary) sick(mary, D) D  measles epidemic(measles)epidemic(D) D  measles

8. Page 8 FOVE P(hospital(mary) | sick(mary, measles)) = ? sick(mary,measles) hospital(mary) sick(mary, D) D  measles epidemic(D) D  measles

9. Page 9 FOVE hospital(mary) sick(mary, D) D  measles epidemic(D) D  measles P(hospital(mary) | sick(mary, measles)) = ?

10. Page 10 FOVE P(hospital(mary) | sick(mary, measles)) = ? hospital(mary) sick(mary, D) D  measles

11. Page 11 FOVE P(hospital(mary) | sick(mary, measles)) = ? hospital(mary)

12. Page 12 FOVE Previously, two operations: Inversion elimination Counting elimination

13. Page 13 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 *

14. Page 14 Inversion Elimination - Conditions - I Eliminated atom must contain all logical variables in parfactors involved. sick(P,D) epidemic(D)

15. Page 15 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

16. Page 16 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)

17. Page 17 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

18. Page 18 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.

19. Page 19 Inversion Elimination - Conditions - II friends(mary,bob) friends(bob,mary) friends(Y,X) friends(X,Y) friends(bob,mary) … X  Y Eliminated RVs must occur in only one instance of parfactor friends(mary,bob) … Inversion Elimination Not Ok

20. Page 20  e(D)  D1  D2  (e(D 1 ),e(D 2 )) =  e(D)  (0,0) #(0,0) in assignment  (0,1) #(0,1) in assignment  (1,0) #(1,0) in assignment  (1,1) #(1,1) in assignment =  # number of assignments (#)  v1,v2  (v1,v2) #v1,v2 Counting Elimination - A Combinatorial Approach

21. Page 21 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

22. Page 22 Uncovered by Inversion and Counting Eliminating epidemic from   epidemic(Disease1,Region), epidemic(Disease2,Region), donations) No atom with all logical variables, so no Inversion Elimination Shared logical variables, so no Counting Elimination

23. Page 23 Partial Inversion  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 ) Subsumes Inversion elimination, which is particular case where all logical variables are inverted.

24. Page 24 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

25. Page 25 Partial inversion conditions   friends(X,Y), friends(Y,X), sick(X,D), sick(Y,D) ) Cannot partially invert on X,Y because friends(bob,mary) appears in more than one instance of parfactor. But now we do not need all logical variables.

26. Page 26 Second contribution: Lifted MPE In propositional case, MPE done by factors containing MPE of eliminated variables. AB C D

27. Page 27 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.

28. Page 28 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.

29. Page 29 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.

30. Page 30 MPE  0.9A=0,B=1,C=1,D=1 In propositional case, MPE done by factors containing MPE of eliminated variables.

31. Page 31 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))

32. Page 32 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

33. Page 33 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

34. Page 34 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,D2 e(D1)=0, e(D2) = 0, r(D1,D2) = 1 9 12 D1,D2 e(D1)=0, e(D2) = 1, r(D1,D2) = 1 9 15 D1,D2 e(D1)=1, e(D2) = 0, r(D1,D2) = 0 9 25 D1,D2 e(D1)=1, e(D2) = 1, r(D1,D2) = 1 ’)’)

35. Page 35 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 Partial 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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