1 lifted first-order probabilistic inference rodrigo de salvo braz university of illinois at...
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Lifted First-Order Probabilistic Inference
Rodrigo de Salvo Braz
University of Illinois atUrbana-Champaign
withEyal Amir and Dan Roth
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Motivation
We want to be able to do inference with Probabilistic First-order Logic if epidemic(Disease) then sick(Person, Disease)
[0.2, 0.05] if sick(Person, Disease) then hospital(Person)
[0.3, 0.01] sick(bob, measles) or sick(bob, flu) 0.6
Expressiveness of logic Robustness of probabilistic models Goal: Probabilistic Inference at First-order level
(scalability one of main advantages)
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Current approaches - Propositionalization
Exact inference: Knowledge-based construction (Breese, 1992) Probabilistic Logic Programming (Ng & Subrahmanian, 1992) Probabilistic Logic Programming (Ngo and Haddawy, 1995) Probabilistic Relational Models (Friendman et al., 1999) Relational Bayesian networks (Jaeger, 1997) Bayesian Logic Programs (Kersting & DeRaedt, 2001) Probabilistic Abduction (Poole, 1993) MEBN (Laskey, 2004) Markov Logic (Richardson & Domingos, 2004)
Sampling: PRISM (Saito, 1995) BLOG (Milch et al, 2005)
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Unexploited structure
(epidemic(Disease1), epidemic(Disease2))
epidemic(measles)
epidemic(flu)
epidemic(rubella)
…
propositionalized
epidemic(D2)
epidemic(D1)
D1 D2
first-order
as in first-order theorem proving
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This Talk
Representation Inference
Inversion Elimination (IE) (Poole, 2003) Formalization of IE Identification of conditions for IE Counting Elimination
Experiment & Conclusions
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Representation
sick(mary,measles)
hospital(mary)
epidemic(measles) epidemic(flu)
sick(mary,flu)
…
… sick(bob,measles)
hospital(bob)
sick(bob,flu)……
…
… …
… …
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Representation
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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Representation
sick(mary,measles)
hospital(mary)
epidemic(measles) epidemic(flu)
sick(mary,flu)
…
… sick(bob,measles)
hospital(bob)
sick(bob,flu)……
…
… …
… …
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Representation - 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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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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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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Representing structure
sick(P,D)
hospital(P)
epidemic(D)
More intutive More compact Represents structure
explicitly Generalization of graphical
models
Atoms representa set of random
variables
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Making use of structure in Inference
Task: given a condition, what is the marginal probability of a set of random variables?
P(sick(bob, measles) | sick(mary,measles)) = ?
Three approaches plain propositionalization dynamic construction (“smart”
propositionalization) lifted inference
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Inference - Plain Propositionalization
Instantiation of potential function for each instantiation
Lots of redundant computation Lots of unnecessary random variables
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Inference - Dynamic construction
Instantiation of potential function for relevant parts
P(hospital(mary) | sick(mary, measles)) = ?
sick(mary,measles)
hospital(mary)
sick(mary, flu) …
epidemic(measles) epidemic(flu)
sick(mary, rubella)
epidemic(rubella)…
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Inference - Dynamic construction
Instantiation of potential function for relevant parts
P(hospital(mary) | sick(mary, measles)) = ?
sick(mary,measles)
hospital(mary)
sick(mary, flu) …
epidemic(measles) epidemic(flu)
sick(mary, rubella)
epidemic(rubella)…
Much redundancy
still
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Inference - Dynamic construction
Most common approach forexact First-order Probabilistic inference:
Knowledge-based construction (Breese, 1992) Probabilistic Logic Programming (Ng & Subrahmanian,
1992) Probabilistic Logic Programming (Ngo and Haddawy,
1995) Probabilistic Relational Models (Friendman et al., 1999) Relational Bayesian networks (Jaeger, 1997) Bayesian Logic Programs (Kersting & DeRaedt, 2001) MEBN (Laskey, 2004) Markov Logic (Richardson & Domingos, 2004)
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Inference - Lifted inference
Inference on parameterized, or first-order, level;
Performs certain inference steps once for a class of random variables;
Poole (2003) describes a generalized Variable Elimination algorithm which we call Inversion Elimination.
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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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Inference - Inversion Elimination (IE)
P(hospital(mary) | sick(mary, measles)) = ?
hospital(mary)
sick(mary, D)
epidemic(D) = Unification
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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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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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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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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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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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Inference - Inversion Elimination (IE)
P(hospital(mary) | sick(mary, measles)) = ?
hospital(mary)
sick(mary, D)
D measles
D measles
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Inference - Inversion Elimination (IE)
P(hospital(mary) | sick(mary, measles)) = ?
hospital(mary)
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…
Inference - Inversion Elimination (IE)
Does not depend on domain size.
hospital(mary)
sick(mary,measles) sick(mary, flu)
hospital(mary)
sick(mary, D)
epidemic(D)epidemic(measles) epidemic(flu)
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First contribution - Formalization of IE
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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First contribution - Formalization of IE
q(X,Y) X,Y (p(X),q(X,Y))
=q(x1,y1)...q(xn,ym)
(p(x1),q(x1,y1))...(p(xn),q(xn,ym))
= (q(x1,y1) (p(x1),q(x1,y1)))... (q(xn,ym) (p(xn),q(xn,ym)))
=X,Y q(X,Y) (p(X),q(X,Y))
= X,Y (p(X)) = X (p(X))
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First contribution - Formalization of IE
q(X,Y) X,Y (p(X),q(X,Y))
=q(x1,y1)...q(xn,ym)
(p(x1),q(x1,y1))...(p(xn),q(xn,ym))
= (q(x1,y1) (p(x1),q(x1,y1)))... (q(xn,ym) (p(xn),q(xn,ym)))
=X,Y q(X,Y) (p(X),q(X,Y))
= X,Y (p(X)) = X (p(X))
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First contribution - Formalization of IE
By formalizing the problem of Inversion Elimination, we determined conditions for its application: Eliminated atom must contain all logical
variables in parfactors involved; Eliminated atom instances must not
occur together in the same instances of parfactor.
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Inversion Elimination - Limitations - I
Eliminated atom must contain all logical variables in parfactors involved.
sick(P,D)
epidemic(D)
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Inversion Elimination - Limitations - I
Eliminated atom must contain all logical variables in parfactors involved.
sick(P,D)
epidemic(D) epidemic(D)
Ok, contains both P and D
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Inversion Elimination - Limitations - 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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Inversion Elimination - Limitations - 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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Inversion Elimination - Limitations - I
Marginalization by eliminating class p(X):
p(X) X,Y (p(X),q(X,Y))
=p(x1)...p(xn) Y (p(x1),q(x1,Y))... Y (p(xn),q(xn,Y))
= (p(x1) Y (p(x1),q(x1,Y)))... (p(xn) Y (p(xn),q(xn,Y)))
=X p(X) Y (p(X),q(X,Y))
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Inversion Elimination - Limitations - II
Requires eliminated RVs to occur in separate instances of parfactor
…sick(mary, flu)
epidemic(flu)
sick(mary, rubella)
epidemic(rubella)…
sick(mary, D)
epidemic(D)
D measles
InversionElimination
Ok
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Inversion Elimination - Limitations - 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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Inversion Elimination - Limitations - II
e(D) D1D2 (e(D1),e(D2))
=e(d1)...e(dn) (e(d1), e(d2)) ... (e(dn-1),e(dn))
=e(d1) (e(d1), e(d2))... e(dn) (e(dn-1),e(dn))
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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
Second Contribution - Counting Elimination
= ( ) v (v)#v in e(D),D1D2 (from i)|e(D)|
i
|e(D)|
i=0
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Second Contribution - Counting Elimination
Does depend on domain size, but exponentially less so than brute force;
More general than Inversion Elimination, but still has conditions of its own; inter-atom logical variables must be in a
dominance ordering p(X,Y), q(Y,X), r (Y) OK, Y dominates X p(X), q(X,Y), r (Y) not OK, no dominance
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A Simple Experiment
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Conclusions
Contributions: Formalization and Identification of conditions
for Inverse Elimination (Poole); Counting Elimination;
Much faster than propositionalization in certain cases;
Basis for probabilistic theorem proving on the First-order level, as it is already the case with regular logic.
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Future Directions
Approximate inference Parameterized queries
“what is the most likely D such that sick(mary, D)?”
Function symbols sick(motherOf(mary), D) sequence(S, [g, c, t])
Equality MPE, MAP Summer project at Cyc
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The End