1 Lifted First-Order Probabilistic Inference Rodrigo de Salvo Braz University of Illinois at Urbana-Champaign with Eyal Amir and Dan Roth.

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1 Lifted First-Order Probabilistic Inference Rodrigo de Salvo Braz University of Illinois at Urbana-Champaign with Eyal Amir and Dan Roth

Page 2 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)

Page 3 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)

Page 4 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

Page 5 This Talk Representation Inference Inversion Elimination (IE) (Poole, 2003) Formalization of IE Identification of conditions for IE Counting Elimination Experiment & Conclusions

Page 6 Representation sick(mary,measles) hospital(mary) epidemic(measles)epidemic(flu) sick(mary,flu) … … sick(bob,measles) hospital(bob) sick(bob,flu) …… … …… ……

Page 7 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))

Page 8 Representation sick(mary,measles) hospital(mary) epidemic(measles)epidemic(flu) sick(mary,flu) … … sick(bob,measles) hospital(bob) sick(bob,flu) …… … …… ……

Page 9 Representation - Lots of Redundancy! sick(mary,measles) hospital(mary) epidemic(measles)epidemic(flu) sick(mary,flu) … … sick(bob,measles) hospital(bob) sick(bob,flu) …… … …… ……

Page 10 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”

Page 11 Parfactor sick(Person,Disease) epidemic(Disease) 8 Person, Disease   sick(Person,Disease), epidemic(Disease))

Page 12 Parfactor sick(Person,Disease) epidemic(Disease) 8 Person, Disease   sick(Person,Disease), epidemic(Disease)), Person  mary, Disease  flu

Page 13 Representing structure sick(P,D) hospital(P) epidemic(D) More intutive More compact Represents structure explicitly Generalization of graphical models Atoms represent a set of random variables

Page 14 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

Page 15 Inference - Plain Propositionalization Instantiation of potential function for each instantiation Lots of redundant computation Lots of unnecessary random variables

Page 16 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) …

Page 17 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

Page 18 Inference - Dynamic construction Most common approach for exact 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)

Page 19 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.

Page 20 Inference - Inversion Elimination (IE) P(hospital(mary) | sick(mary, measles)) = ? hospital(mary) sick(mary, D) epidemic(D)

Page 21 Inference - Inversion Elimination (IE) P(hospital(mary) | sick(mary, measles)) = ? hospital(mary) sick(mary, D) epidemic(D) = Unification

Page 22 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

Page 23 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 =

Page 24 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

Page 25 Inference - Inversion Elimination (IE) P(hospital(mary) | sick(mary, measles)) = ? sick(mary,measles) hospital(mary) sick(mary, D) D  measles epidemic(D) D  measles

Page 26 Inference - Inversion Elimination (IE) hospital(mary) sick(mary, D) D  measles epidemic(D) D  measles P(hospital(mary) | sick(mary, measles)) = ?

Page 27 Inference - Inversion Elimination (IE) P(hospital(mary) | sick(mary, measles)) = ? hospital(mary) sick(mary, D) D  measles

Page 28 Inference - Inversion Elimination (IE) P(hospital(mary) | sick(mary, measles)) = ? hospital(mary)

Page 29 … 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)

Page 30 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))

Page 31 First contribution - Formalization of IE  q(X,Y)  X,Y   (p(X),q(X,Y)) =  q(x1,y1)...  q(xn,ym)   (p(x 1 ),q(x 1,y 1 ))...   (p(x n ),q(x n,y m )) = (  q(x1,y1)   (p(x 1 ),q(x 1,y 1 )) )... (  q(xn,ym)   (p(x n ),q(x n,y m )) ) =  X,Y  q(X,Y)   (p(X),q(X,Y)) =  X,Y   (p(X)) =  X   (p(X))

Page 32 First contribution - Formalization of IE  q(X,Y)  X,Y   (p(X),q(X,Y)) =  q(x1,y1)...  q(xn,ym)   (p(x 1 ),q(x 1,y 1 ))...   (p(x n ),q(x n,y m )) = (  q(x1,y1)   (p(x 1 ),q(x 1,y 1 )) )... (  q(xn,ym)   (p(x n ),q(x n,y m )) ) =  X,Y  q(X,Y)   (p(X),q(X,Y)) =  X,Y   (p(X)) =  X   (p(X))

Page 33 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.

Page 34 Inversion Elimination - Limitations - I Eliminated atom must contain all logical variables in parfactors involved. sick(P,D) epidemic(D)

Page 35 Inversion Elimination - Limitations - I Eliminated atom must contain all logical variables in parfactors involved. sick(P,D) epidemic(D) Ok, contains both P and D

Page 36 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)

Page 37 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

Page 38 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(x 1 ),q(x 1,Y))...  Y   (p(x n ),q(x n,Y)) = (  p(x1)  Y   (p(x 1 ),q(x 1,Y)) )... (  p(xn)  Y   (p(x n ),q(x n,Y)) ) =  X  p(X)  Y   (p(X),q(X,Y))

Page 39 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 Inversion Elimination Ok

Page 40 Inversion Elimination - Limitations - II epidemic(measles) epidemic(flu) epidemic(D2) epidemic(D1) epidemic(rubella) … Inversion Elimination Not Ok D1  D2 Requires eliminated RVs to occur in separate instances of parfactor

Page 41 Inversion Elimination - Limitations - II  e(D)  D1  D2  (e(D 1 ),e(D 2 )) =  e(d1)...  e(dn)  (e(d 1 ), e(d 2 ))...  (e(d n-1 ),e(d n )) =  e(d1)  (e(d 1 ), e(d 2 ))...  e(dn)  (e(d n-1 ),e(d n ))

Page 42  e(D)  D1  D2  (e(D 1 ),e(D 2 )) =  e(D)  (0,0) #(0,0) in e(D),D1  D2  (0,1) #(0,1) in e(D),D1  D2  (1,0) #(1,0) in e(D),D1  D2  (1,1) #(1,1) in e(D),D1  D2 =  e(D)  v  (v) #v in e(D),D1  D2 Second Contribution - Counting Elimination =  ( )  v  (v) #v in e(D),D1  D2 (from i) |e(D)| i i=0

Page 43 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

Page 44 A Simple Experiment

Page 45 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.

Page 46 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

Page 47 The End