MPE and Partial Inversion in Lifted Probabilistic Variable Elimination

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

Lifted Probabilistic Inference • We assume probabilistic statements such as8Person, DiseaseP(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

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

epidemic(Disease) sick(Person,Disease) Parfactor 8 Person, Disease f(sick(Person,Disease), epidemic(Disease))

epidemic(Disease) sick(Person,Disease) Parfactor Person  mary, Disease  flu 8 Person, Disease f(sick(Person,Disease), epidemic(Disease)), Person  mary, Disease  flu

Joint Distribution • As in propositional case, proportional to product of all factors • But here, “all factors” means all instantiations of all parfactors: P(...) ÕXf1(p(X)) ÕX,Yf2(p(X),q(X,Y))

Inference task - Marginalization åq(X,Y)ÕXf1(p(X)) ÕX,Yf2(p(X),q(X,Y)) Marginal on all random variables in p(X): summation over all assignments to all instances of q(X,Y)

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

FOVE P(hospital(mary)) = ? epidemic(measles) epidemic(D) D  measles sick(mary,measles) sick(mary, D) D  measles hospital(mary)

FOVE P(hospital(mary)) = ? epidemic(D) D  measles sick(mary,measles) sick(mary, D) D  measles hospital(mary)

FOVE P(hospital(mary)) = ? epidemic(D) D  measles sick(mary, D) D  measles hospital(mary)

FOVE P(hospital(mary)) = ? D  measles sick(mary, D) D  measles hospital(mary)

FOVE P(hospital(mary)) = ? hospital(mary)

Counting Elimination - A Combinatorial Approach åe(D)ÕD1D2f(e(D1),e(D2)) = åe(D)f(0,0)#(0,0) in assignment f(0,1)#(0,1) in assignment f(1,0)#(1,0) in assignment f(1,1)#(1,1) in assignment Let i be the number of e(D)’s assigned 1: = åi Õv1,v2f(v1,v2)#(v1,v2) given i (number of assignments with |{D : e(D)=1}| = i)

Counting Elimination - Conditions • It does not work oneliminating class epidemic fromf(epidemic(D1, Region), epidemic(D2, Region), donations). • In general, counting elimination does not apply when atoms share logical variables. • Here, Region is shared between atoms.

Partial Inversion Provides a way of not sharing logical variables åe(D,R) ÕD1D2,R f( e(D1,R), e(D2,R), d ) = ÕR åe(D,r) ÕD1D2 f( e(D1,r), e(D2,r), d ) (R is now bound, so not a variable anymore) = ÕR f’( d ) = f’( d )|R|= f’’( d )

Partial Inversion, graphically epidemic(D1,R) Each instance a counting elimination problem donations D1  D2 epidemic(D2,R) … epidemic(D1,r1) epidemic(D1,r10) … D1  D2 D1  D2 epidemic(D2,r1) epidemic(D2,r10) donations

Another (not so partial) inversion åq(X,Y)ÕX,Yf(p(X),q(X,Y)) (expensive) =ÕX,Yåq(X,Y) f(p(X),q(X,Y)) (propositional) = ÕX,Yf'(p(X)) = ÕXf'|Y|(p(X)) = ÕXf''(p(X)) (marginal on p(X))

Another (not so partial) inversion p(X) Each instance a propositional elimination problem q(X,Y) … p(x1) p(xn) … q(x1,y1) q(xn,yn)

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

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

Second contribution: Lifted MPE • In propositional case,MPE done by factors containing MPE of eliminated variables. C A B D

MPE • In propositional case,MPE done by factors containing MPE of eliminated variables. A B D

MPE • In propositional case,MPE done by factors containing MPE of eliminated variables. A B

MPE • In propositional case,MPE done by factors containing MPE of eliminated variables. A

MPE • In propositional case,MPE done by factors containing MPE of eliminated variables.

MPE • Same idea in First-order case • But factors are quantified and so are assignments: 8 X, Y f(p(X), q(X,Y))

MPE 8 X, Y f(p(X), q(X,Y)) After Inversion Elimination of q(X,Y): Liftedassignments 8 X f’(p(X))

MPE After Inversion Elimination of p(X): 8 X f’(p(X)) f’’()

MPE 8 D1, D2 f(e(D1), e(D2)) After Counting Elimination of e: f’()

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.

MPE and Partial Inversion in Lifted Probabilistic Variable Elimination