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

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)

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)

Unexploited structure

(epidemic(Disease1), epidemic(Disease2))

epidemic(measles)

epidemic(flu)

epidemic(rubella)

propositionalized

epidemic(D2)

epidemic(D1)

first-order

as in first-order theorem proving

This Talk

Representation Inference

Inversion Elimination (IE) (Poole, 2003) Formalization of IE Identification of conditions for IE Counting Elimination

Experiment & Conclusions

Representation

sick(mary,measles)

hospital(mary)

epidemic(measles) epidemic(flu)

sick(mary,flu)

… sick(bob,measles)

hospital(bob)

sick(bob,flu)……

… …

Representation

sick(mary,measles)

hospital(mary)

sick(mary,flu)

hospital(bob)

sick(bob,flu)……

… …

… …sick(mary,measles),

epidemic(measles))

Representation

sick(mary,measles)

hospital(mary)

sick(mary,flu)

hospital(bob)

sick(bob,flu)……

… …

Representation - Lots of Redundancy!

sick(mary,measles)

hospital(mary)

sick(mary,flu)

hospital(bob)

sick(bob,flu)……

… …

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”

Parfactor

sick(Person,Disease)

epidemic(Disease)

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

Parfactor

sick(Person,Disease)

epidemic(Disease)

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

Person mary, Disease flu

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

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

Inference - Plain Propositionalization

Instantiation of potential function for each instantiation

Lots of redundant computation Lots of unnecessary random variables

Inference - Dynamic construction

Instantiation of potential function for relevant parts

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

sick(mary,measles)

hospital(mary)

sick(mary, flu) …

sick(mary, rubella)

epidemic(rubella)…

Instantiation of potential function for relevant parts

sick(mary,measles)

hospital(mary)

sick(mary, flu) …

sick(mary, rubella)

Much redundancy

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)

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.

Inference - Inversion Elimination (IE)

hospital(mary)

sick(mary, D)

epidemic(D)

hospital(mary)

sick(mary, D)

epidemic(D) = Unification

sick(mary,measles)

hospital(mary)

sick(mary, D)

D measles

epidemic(measles) epidemic(D)

D measles

sick(mary,measles)

hospital(mary)

sick(mary, D)

D measles

D measles=

sick(mary,measles)

hospital(mary)

sick(mary, D)

D measles

sick(mary,measles)

hospital(mary)

sick(mary, D)

D measles

epidemic(D)

D measles

hospital(mary)

sick(mary, D)

D measles

epidemic(D)

D measles

hospital(mary)

sick(mary, D)

D measles

hospital(mary)

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)

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))

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))

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))

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.

Inversion Elimination - Limitations - I

Eliminated atom must contain all logical variables in parfactors involved.

sick(P,D)

epidemic(D)

sick(P,D)

epidemic(D) epidemic(D)

Ok, contains both P and D

sick(P,D)

epidemic(D)

Not Ok, missing P

sick(P,D)

q(Y,Z)

p(X,Y)No atom can be

eliminated

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))

Inversion Elimination - Limitations - II

Requires eliminated RVs to occur in separate instances of parfactor

…sick(mary, flu)

epidemic(flu)

sick(mary, rubella)

sick(mary, D)

epidemic(D)

D measles

InversionElimination

epidemic(measles)

epidemic(flu)

epidemic(D2)

epidemic(D1)

epidemic(rubella)

InversionElimination

Not OkD1 D2

Requires eliminated RVs to occur in separate instances of parfactor

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))

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)|

|e(D)|

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

A Simple Experiment

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.

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

The End

1 Lifted First-Order Probabilistic Inference Rodrigo de Salvo Braz University of Illinois at...

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