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

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) …… … …… ……

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))

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) …… … …… ……

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) …… … …… ……

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”

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

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

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.

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

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

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

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 =

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

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

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

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

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

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))

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 *

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

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

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)

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

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.

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

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

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

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

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.

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

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.

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

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

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.

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.

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

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))

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

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

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 ’)’)

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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