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MPE and Partial Inversion inLifted Probabilistic Variable Elimination

Rodrigo de Salvo Braz

University of Illinois at

Urbana-Champaignwith

Eyal Amir and Dan Roth

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Lifted Probabilistic Inference

We assume probabilistic statements such as8 Person, 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

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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 variabl

e

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Parfactor

sick(Person,Disease)

epidemic(Disease)

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

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Parfactor

sick(Person,Disease)

epidemic(Disease)

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

Person mary, Disease flu

Person mary, Disease flu

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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))

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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)

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The FOVE Algorithm

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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FOVE

P(hospital(mary)) = ?

sick(mary,measles)

hospital(mary)

sick(mary, D)

D measles

epidemic(measles) epidemic(D)

D measles

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FOVE

P(hospital(mary)) = ?

sick(mary,measles)

hospital(mary)

sick(mary, D)

D measles

epidemic(D)

D measles

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FOVE

hospital(mary)

sick(mary, D)

D measles

epidemic(D)

D measles

P(hospital(mary)) = ?

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FOVE

P(hospital(mary)) = ?

hospital(mary)

sick(mary, D)

D measles

D measles

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FOVE

P(hospital(mary)) = ?

hospital(mary)

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e(D) D1D2 (e(D1),e(D2))

= 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

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It does not work oneliminating class epidemic from(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.

Counting Elimination - Conditions

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Partial Inversion

Provides a way of not sharing logical variables

e(D,R) D1D2,R e(D1,R), e(D2,R), d )

R e(D,r) D1D2 e(D1,r), e(D2,r), d )

(R is now bound, so not a variable anymore)

R ’d ) = ’d )|R| = ’’d )

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Partial Inversion, graphically

epidemic(D2,r1)

epidemic(D1,r1)

D1 D2

donations

epidemic(D2,R)

epidemic(D1,R)

D1 D2 donations

epidemic(D2,r10)

epidemic(D1,r10)

D1 D2…

…

Each instance a counting

elimination problem

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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))

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Another (not so partial) inversion

…q(x1,y1)

p(x1)

q(xn,yn)

p(xn)…

q(X,Y)

p(X)Each instance a

propositional elimination

problem

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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)

…

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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.

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Second contribution: Lifted MPE

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

A B

C

D

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MPE

A B

D

B D MPE

0 0 0.3 C=1

0 1 0.2 C=1

1 0 0.5 C=0

1 1 0.9 C=1

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

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MPE

A B

B MPE

0 0.5 C=1,D=0

1 1.4 C=1,D=1

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

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MPE

A

A MPE(B,C,D)

0 0.9 B=0,C=1,D=0

1 0.7 B=1,C=1,D=1

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

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MPE

MPE

0.9 A=0,B=1,C=1,D=1

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

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MPE

Same idea in First-order case But factors are quantified and so are assignments:

p(X) q(X,Y) MPE

0 0 0.3 r(X,Y) = 1

0 1 0.2 r(X,Y) = 1

1 0 0.5 r(X,Y) = 0

1 1 0.9 r(X,Y) = 1

8 X, Y (p(X), q(X,Y))

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MPE

After Inversion Elimination of q(X,Y):

p(X) q(X,Y) MPE

0 0 0.3 r(X,Y) = 1

0 1 0.9 r(X,Y) = 1

1 0 0.5 r(X,Y) = 0

1 1 0.3 r(X,Y) = 1

8 X, Y (p(X), q(X,Y))

p(X) ’ MPE

0 0.05 8 Y q(X,Y) = 1, r(X,Y) = 1

1 0.02 8 Y q(X,Y) = 0, r(X,Y) = 1

8 X ’(p(X))

Liftedassignment

s

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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

0 0.05 8 Y q(X,Y) = 1, r(X,Y) = 1

1 0.02 8 Y q(X,Y) = 0, r(X,Y) = 1

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MPE

After Counting Elimination of e:

e(D1) e(D2) MPE

0 0 0.3 r(D1,D2) = 1

0 1 0.9 r(D1,D2) = 1

1 0 0.5 r(D1,D2) = 0

1 1 0.3 r(D1,D2) = 1

8 D1, D2 (e(D1), e(D2))

’ MPE

0.05 938 D1,D2 e(D1)=0, e(D2) = 0, r(D1,D2) = 1

912 D1,D2 e(D1)=0, e(D2) = 1, r(D1,D2) = 1

915 D1,D2 e(D1)=1, e(D2) = 0, r(D1,D2) = 0

925 D1,D2 e(D1)=1, e(D2) = 1, r(D1,D2) = 1

’()

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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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