1 MPE and Partial Inversion in Lifted Probabilistic Variable Elimination Rodrigo de Salvo Braz University of Illinois at Urbana-Champaign with Eyal Amir

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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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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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sick(mary,measles)

hospital(mary)


sick(mary,flu)

…


hospital(bob)

sick(bob,flu)……

…

… …

… …sick(mary,measles),

epidemic(measles))

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sick(mary,measles)

hospital(mary)


sick(mary,flu)

…


hospital(bob)

sick(bob,flu)……

…

… …

… …

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Lots of Redundancy!

sick(mary,measles)

hospital(mary)


sick(mary,flu)

…


hospital(bob)

sick(bob,flu)……

…

… …

… …

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Representing structure

sick(mary,measles)


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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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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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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Inference - Inversion Elimination (IE)

P(hospital(mary) | sick(mary, measles)) = ?

hospital(mary)

sick(mary, D)

epidemic(D)

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hospital(mary)

sick(mary, D)

epidemic(D) = Unification

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sick(mary,measles)

hospital(mary)

sick(mary, D)

D measles

epidemic(measles) epidemic(D)

D measles

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sick(mary,measles)

hospital(mary)

sick(mary, D)

D measles


D measles=

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sick(mary,measles)

hospital(mary)

sick(mary, D)

D measles


D measles

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sick(mary,measles)

hospital(mary)

sick(mary, D)

D measles

epidemic(D)

D measles

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hospital(mary)

sick(mary, D)

D measles

epidemic(D)

D measles


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hospital(mary)

sick(mary, D)

D measles

D measles

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hospital(mary)

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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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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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Inversion Elimination - Conditions - I

Eliminated atom must contain all logical variables in parfactors involved.

sick(P,D)

epidemic(D)

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sick(P,D)

epidemic(D) epidemic(D)

Ok, contains both P and D

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sick(P,D)

epidemic(D)

Not Ok, missing P

sick(P,D)

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q(Y,Z)

p(X,Y)No atom can be

eliminated

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…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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Inversion Elimination - Conditions - II

epidemic(measles)

epidemic(flu)

epidemic(D2)

epidemic(D1)

epidemic(rubella)

…

InversionElimination

Not OkD1 D2

Requires eliminated RVs to occur in separate instances of parfactor

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

= e(D) (0,0)#(0,0) in e(D),D1D2

(0,1)#(0,1) in e(D),D1D2

(1,0)#(1,0) in e(D),D1D2

(1,1)#(1,1) in e(D),D1D2

= e(D) v (v)#v in e(D),D1D2

Counting Elimination - A Combinatorial Approach

= ( ) v (v)#v in e(D),D1D2 (from i)|e(D)|

i

|e(D)|

i=0

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No shared logical variables between atoms,so counting can be done independently

(epidemic(D1, Region), epidemic(D2, Region))

Counting Elimination - A Combinatorial Approach

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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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Partial Inversion

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

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 e(D,r) D1D2 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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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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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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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


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MPE

A B

B MPE

0 0.5 C=1,D=0

1 1.4 C=1,D=1


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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


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MPE

MPE

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


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

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

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

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

’)

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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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Documents

1 MPE and Partial Inversion in Lifted Probabilistic Variable Elimination Rodrigo de Salvo Braz University of Illinois at Urbana-Champaign with Eyal Amir