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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

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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) …… … …… ……

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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))

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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) …… … …… ……

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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) …… … …… ……

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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”

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Page 7 Parfactor sick(Person,Disease) epidemic(Disease) 8 Person, Disease sick(Person,Disease), epidemic(Disease))

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Page 8 Parfactor sick(Person,Disease) epidemic(Disease) 8 Person, Disease sick(Person,Disease), epidemic(Disease)), Person mary, Disease flu

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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.

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

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

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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

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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 =

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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

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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

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

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

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

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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))

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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 *

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

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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

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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)

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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

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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.

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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

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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

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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

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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

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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.

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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

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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.

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

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

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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.

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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.

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

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Page 38 MPE Same idea in First-order case But factors are quantified and so are assignments: p(X)q(X,Y) MPE r(X,Y) = r(X,Y) = r(X,Y) = r(X,Y) = 1 8 X, Y p(X), q(X,Y))

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Page 39 MPE After Inversion Elimination of q(X,Y): p(X)q(X,Y) MPE r(X,Y) = r(X,Y) = r(X,Y) = r(X,Y) = 1 8 X, Y p(X), q(X,Y)) p(X) ’’ MPE Y q(X,Y) = 1, r(X,Y) = Y q(X,Y) = 0, r(X,Y) = 1 8 X ’ p(X)) Lifted assignments

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Page 40 MPE After Inversion Elimination of p(X): 8 X ’ p(X)) ’’ MPE X 8 Y p(X) = 0, q(X,Y) = 1, r(X,Y) = 0 ’’ ) p(X) ’’ MPE Y q(X,Y) = 1, r(X,Y) = Y q(X,Y) = 0, r(X,Y) = 1

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Page 41 MPE After Counting Elimination of e: e(D1)e(D2) MPE r(D1,D2) = r(D1,D2) = r(D1,D2) = r(D1,D2) = 1 8 D1, D2 e(D1), e(D2)) ’’ MPE (D1=0,D2=0) e(D1)=0, e(D2) = 1, r(D1,D2) = (D1=0,D2=1) e(D1)=1, e(D2) = 1, r(D1,D2) = (D1=1,D2=0) e(D1)=1, e(D2) = 0, r(D1,D2) = (D1=1,D2=1) e(D1)=0, e(D2) = 0, r(D1,D2) = 1 ’)’)

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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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