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 Lifted Probabilistic Inference We assume probabilistic statements such as 8 Person, Disease P(sick(Person,Disease) | epidemics(Disease)) = 0.3 Typical approach is grounding. We seek to do inference at first-order level, like it is done in logic. Faster and more intelligible. Two contributions: Partial inversion: more general technique than 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 Joint Distribution As in propositional case, proportional to product of all factors But here, all factors means all instantiations of all parfactors: P(...) X (p(X)) X,Y (p(X),q(X,Y))

7. Page 7 Inference task - Marginalization q(X,Y) X (p(X)) X,Y (p(X),q(X,Y)) Marginal on all random variables in p(X): summation over all assignments to all instances of q(X,Y)

8. Page 8 The FOVE Algorithm First-Order Variable Elimination (FOVE): a generalization of Variable Elimination in propositional graphical models. Eliminates classes of random variables at once.

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

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

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

12. Page 12 FOVE P(hospital(mary)) = ? hospital(mary) sick(mary, D) D measles

13. Page 13 FOVE P(hospital(mary)) = ? hospital(mary)

14. Page 14 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 Let i be the number of e(D)s assigned 1: = i v1,v2 (v1,v2) #(v1,v2) given i (number of assignments with |{D : e(D)=1}| = i) Counting Elimination - A Combinatorial Approach

15. Page 15 It does not work on eliminating class epidemic from (epidemic(D 1, Region), epidemic(D 2, Region), donations). In general, counting elimination does not apply when atoms share logical variables. Here, Region is shared between atoms. Counting Elimination - Conditions

16. Page 16 Partial Inversion Provides a way of not sharing logical variables 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 is now bound, so not a variable anymore) R d ) = d ) |R| = d )

17. Page 17 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

18. Page 18 Another (not so partial) inversion q(X,Y) X,Y (p(X),q(X,Y)) (expensive) = X,Y q(X,Y) (p(X),q(X,Y)) (propositional) = X,Y '(p(X)) = X ' Y (p(X)) = X ''(p(X)) (marginal on p(X) )

19. Page 19 Another (not so partial) inversion … q(x 1,y 1 ) p(x 1 ) q(x n,y n ) p(x n ) … q(X,Y) p(X) Each instance a propositional elimination problem

20. Page 20 Partial inversion conditions friends(X,Y), friends(Y,X)) Cannot partially invert on X,Y because friends(bob,mary) appears in more than one instance of parfactor. friends(mary,bob) friends(bob,mary) friends(Y,X) friends(X,Y) friends(bob,mary) … X Y friends(mary,bob) …

21. Page 21 Summary of Partial Inversion More general than previous Inversion Elimination. Generates Counting Elimination or Propositional sub-problems. Cannot be applied to entangled parfactors. Does not depend on domain size.

22. Page 22 Second contribution: Lifted MPE In propositional case, MPE done by factors containing MPE of eliminated variables. AB C D

23. Page 23 MPE AB D BD MPE 000.3C=1 010.2C=1 100.5C=0 110.9C=1 In propositional case, MPE done by factors containing MPE of eliminated variables.

24. Page 24 MPE AB B MPE 00.5C=1,D=0 11.4C=1,D=1 In propositional case, MPE done by factors containing MPE of eliminated variables.

25. Page 25 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.

26. Page 26 MPE MPE 0.9A=0,B=1,C=1,D=1 In propositional case, MPE done by factors containing MPE of eliminated variables.

27. Page 27 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))

28. Page 28 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

29. Page 29 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

30. Page 30 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 ()

31. Page 31 Conclusions Partial Inversion: More general algorithm, subsumes Inversion elimination Lifted Most Probable Explanation (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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