CS498EA Reasoning in AI Lecture

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CS498-EAReasoning in AILecture 25

• Instructor Eyal Amir
• Fall Semester 2009

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What is Reasoning in AI?
Combining Logic and Probability
Logical Reasoning
Probabilistic Reasoning
Commonsense Reasoning
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Previously Combining Models of Logic and
Probability

• Representation methods for models that combine
  relationships between objects and probabilities
• Relational Markov Networks
• Probabilistic Relational Models
• Markov Logic Networks
• Many others
• If Have time
• PRMs w/ Attribute Uncertainty
• PRMs w/ Link Uncertainty

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Example A Recession Model

• What is probability of recession, when a bank(bm)
  goes into bankruptcy?
• Recession Recession of a country in 0,1
• MarketX Quarterly market (X) index
• LossX,Y Loss of a bank (Y) in a market (X)
• RevenueY Revenue of a bank (Y)

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Social Networks

• Example school friendships and their effects

Friend(A,B)
Attribute(A)
Measuremt(A)
Friend(A,C)
Attribute(B)
Measuremt(B)
Friend(B,C)
Attribute(C)
Measuremt(C)
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Modeling Epidemics
epidemic(measles)
epidemic(flu)



sick(mary,measles)
sick(mary,flu)
sick(bob,measles)
sick(bob,flu)





hospital(mary)
hospital(bob)

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Patterns in Structured Data
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Probabilistic Relational Model
Strain
Patient
Unique
POB
Homeless
HIV-Result
Contact
Age
Disease Site
Contact-Type
Close-Contact
Transmitted
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PRM Semantics
Contact c1
Strain s1
Patient p2
Contact c2
Strain s2
Patient p1
Contact c3
Patient p3
PRM
relational skeleton ?


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Representation Languages

• Many developed 1990s-2000s
• Probabilistic Logic (Nilsson 1986)
• First-Order probabilistic logics (Joe Halpern
  1990)
• Relational Markov Networks (Zhang Poole 1993)
• Relational Probabilistic Models (Pfeffer Koller
  1997)
• Knowledge-based construction (Breese, 1992)
• Probabilistic Logic Programs (Ng Subrahmanian,
  1992)
• Stochastic Logic Programs (Ngo Haddawy, 1995)
• Bayesian Logic Programs (Kersting DeRaedt,
  2001)
• Relational Dependency Networks (Neville Jensen,
  2004)
• Relational Bayesian networks (Jaeger, 1997)
• MEBN (Laskey, 2004)
• Markov Logic Networks (Richardson Domingos,
  2004)
• BLOG Bayesian LOGic (Milch etal. 2004)

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Representation Languages

• Relational Markov Networks (Zhang Poole 1993)
• Probabilistic Relational Models (Koller Pfeffer
  1997)
• Markov Logic Networks (Richardson Domingos,
  2004)
• What they can represent
• Single probabilistic model
• Distribution over an unknown (but finite) number
  of elements
• Unique Names assumed for elements
• Hard to represent (and to reason with tractably)
• Uncertainty about equality of elements
• Functions over elements
• .

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Useful For

• Representing high-dimensional distributions
• Natural-Language Processing
• Social Network Analysis
• Stochastic Relational Databases
• Not so useful yet
• Scaling up applications to many elements
• Needs efficient, precise inference

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Today

• Inference in PRMS / RMNs/ MLNs
• Sampling
• Lifted Inference

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Inference in Unrolled BN

• Exact Inference in unrolled BN
• Infeasible for large networks
• Structural (Attr/Reference/Exists) Uncertainty
  creates very large cliques
• Use caching (Pfeffer 00)
• FOL-Resolution-style techniques
• Loopy belief propagation (Pearl, 88 McEliece,
  98)
• Scales linearly with size of network
• Guaranteed to converge only for polytrees
• Empirically, often converges in general nets
  (Murphy99)
• Use approx. inference MCMC (Pasula etal. 01)

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MCMC with PRMs
Prof1.
Prof2.
Prof3.
Prof1. fame
Prof2. fame
Prof3. fame
Student1. advisor
Student1. success
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MCMC with PRMs
Prof1.
Prof2.
Prof3.
Network structure changed
Prof2. fame
Student1. advisor
Prof2
Student1. success
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Gibbs Sampling with PRMs

• For each complex attribute A reference attribute
  RefA, w/finite domain ValRefA
• Reference uncertainty modifies chain of
  attributes

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Gibbs Sampling with PRMs

• For each complex attribute A reference attribute
  RefA, w/finite domain ValRefA
• Reference uncertainty modifies chain of
  attributes
• Gibbs for simple attributes Use MB
• Gibbs for complex attributes (RU)
• Add reference variables

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Gibbs Sampling with PRMs
Gibbs when reference var does not change
Prof1.
Prof2.
Prof3.
Prof2. fame
Student1. advisor
Prof3. fame
Prof2
Student1. success
P(P3.f mb(P3.f)) ?P(P3.fPa(P3.f))P(P3.P3.f)
P(S1.sS1.aP2,P1.f,P2.f,P3.f) ?P(P3.f) P(P3.
P3.f) P(S1.s S1.aP2,P2.f) ?P(P3.f) P(P3.
P3.f)
Constant wrt P3.f
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M-H Sampling with PRMs
Changing a ref. variable
Prof1.
Prof2.
Prof3.
Prof2. fame
Student1. advisor
Prof3. fame
Prof2
Student1. success
P(s1.aP3,...X) q(s1.aP2,...X s1.aP3,...X)
-------------------------------------------------
-------------------- P(s1.aP2,...X)
q(s1.aP3,...X s1.aP2,...X)
P(s1.aP3,...X) P(s1.aP3 P1.,,Pn.)
P(s1.sP3.f, ------------------------
----------------------------------------- P(s1.aP
2,...X) P(s1.aP3,...X)
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M-H Sampling with PRMs
Changing a ref. variable
Prof1.
Prof2.
Prof3.
Prof2. fame
Student1. advisor
Prof3. fame
Prof2
Student1. success
P(s1.aP3,...X) ------------------------
P(s1.aP2,...X)
P(s1.aP3 P1.,,Pn.) P(s1.s
P3.f,S1.aP3) ------------------------------------
------------------------------- P(s1.aP2
P1.,,Pn.) P(s1.s P2.f,S1.aP2)
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M-H Sampling with PRMs
Changing a ref. variable
Prof1.
Prof2.
Prof3.
Prof2. fame
Student1. advisor
Prof3. fame
When aggregation function (e.g.,max,
softmax)
Prof2
Student1. success
P(s1.aP3 P1.,,Pn.) P(s1.s
P3.f,S1.aP3) ------------------------------------
-------------------------------- P(s1.aP2
P1.,,Pn.) P(s1.s P2.f,S1.aP2)
P(s1.aP3 P3.) P(s1.s P3.f,S1.aP3) --------
------------------------------------------------ P
(s1.aP2 P2.) P(s1.s P2.f,S1.aP2)
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Today

• Inference in PRMS / RMNs/ MLNs
• Sampling
• Lifted Inference

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Example Inference Problem
epidemic(measles)
epidemic(flu)



Problem Inference exponential in vars!
sick(mary,measles)
sick(mary,flu)
sick(bob,measles)
sick(bob,flu)





hospital(mary)
hospital(bob)

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Making use of structure in Inference

• Task calculate marginals and posteriorsP(sick(b
  ob, measles) sick(mary,measles)) ?
• Three approaches
• plain propositionalization
• dynamic construction (smart propositionalization
  )
• Our focus today lifted inference

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Lifted Variable EliminationHigh-Level View
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Step 1 Inversion Elimination

• Joint distribution
• ÕP,D f1(e(D),s(P,D)) f2(s(P,D),h(P))
• Marginalization by eliminating class s(P,D)
• ås(.,.) ÕP,D f1(e(D),s(P,D)) f2(s(P,D),h(P))

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Step 1 Inversion Elimination

• ås(.,.) ÕP,D f1(e(D),s(P,D))f2(s(P,D),h(P))
• ÕP,D ås(P,D) f1(e(D),s(P,D))f2(s(P,D),h(P))
• ÕP,D f3(e(D),h(P))

Important computing??3(X,Y) is independent of
D,P ??3(X,Y) åZ f1(X,Z) f2(Z,Y)
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Inversion Elimination

• ås(.,.) ÕP,D f1(e(D),s(P,D)) f2(s(P,D),h(P))
• ås(p1,d1)ås(p1,d2)...ås(pn,dm)
  f1(e(d1),s(p1,d1))?f1(e(d1),s(p1,d2))...f2(s(p1,d
  1),h(p1))?f2(s(p1,d2),h(p1))...
• (ås(p1,d1) f1(e(d1),s(p1,d1))f2(s(p1,d1),h(p1)))
  (ås(p1,d2) f1(e(d1),s(p1,d2))f2(s(p1,d2),h(p1)))
  ...(ås(pn,dm) f1(e(dn),s(pn,dm))f2(s(pn,dm),h(pn)
  ))
• ÕP,D ås(P,D) f1(e(D),s(P,D)) f2(s(P,D),h(P))

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Counting Elimination

• åe(.)h(.) ÕD,P f3(e(D),h(P))
• åe(.)h(.) f3(0,0)(0,0) in e(.),h(.)
  f3(0,1)(0,1) in e(.),h(.)
  f3(1,0)(1,0) in e(.),h(.)
  f3(1,1)(1,1) in e(.),h(.)
• åe(.)h(.) Õv f3(v)v in e(.),h(.)

Does depend on domain size, but not exponentially
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Experimental Comparison with Propositional
Inference Methods
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Relational Factoring ofSocial Network Analysis

• Example school friendships and their effects

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Scaling-Up Experiments Computing Pr(friend(x,y))
Figure 5 Computation time for
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Reasoning A Recession Model

• What is probability of recession, when a bank(bm)
  goes into bankruptcy?
• Recession Recession of a country in 0,1
• MarketX Quarterly market (X) index
• LossX,Y Loss of a bank (Y) in a market (X)
• RevenueY Revenue of a bank (Y)

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Experiments
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Conclusions

• Lifted inference with discrete variables holds
  promise for scaling up of models and inference
• Sampling in relational models allows inferences
  about individuals while taking into account
  relational structure (only correct independence
  assumptions)
• Social Network Analysis feasible application
• Can infer unobserved attributes from partial
  friendship and attribute data
• Can infer uncertain friendship links from partial
  friendship and attribute data
• Open Questions
• Friendship structures and transitivity
• MPE Most Probable Explanations
• Identity uncertainty

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Challenges

• Parameterized queries P(sick(A, measles)) ?
• Function symbols ?(diabetes(A),
  diabetes(father(A)))
• Equality AB
• Transitivity-like potential functions
• ?(friend(A,B),friend(B,C),friend(A,C))
• Observations
• seen(john)1, seen(james)0
• in_hospital(john)1, in_hospital(james)1
• Continuous and Hybrid relational models
• Economic models
• Switching Kalman Filters

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References

• R. D. S. Braz, Lifted first-order probabilistic
  inference, PhD thesis, University of Illinois at
  Urbana-Champaign, 2007.
• R. D. S. Braz, E. Amir, and D. Roth, Lifted
  first-order probabilistic inference, in 19th
  Intl' Joint Conference on Artificial Intelligence
  (IJCAI'05), AAAI Press, 2005, pp. 1319-1325.
• R. D. S. Braz, E. Amir, and D. Roth, Mpe and
  partial inversion in lifted probabilistic
  variable elimination, in 21st National Conference
  on Artificial Intelligence (AAAI'06), AAAI Press,
  2006, pp. 1123-1130.
• R. D. S. Braz, E. Amir, and D. Roth, Lifted
  first-order probabilistic inference, in Lise
  Getoor and Ben Taskar, eds. Statistical
  Relational Learning, MIT Press, 2007, pp.
  433-452.
• L. Getoor and B. Taskar, Introduction to
  statistical relational learning, MIT Press, 2007.
• B. Milch, L. S. Zettlemoyer, K. Kersting, M.
  Haimes, and L. P. Kaelbling, Lifted probabilistic
  inference with counting formulas, in Proc. 23rd
  AAAI Conference on Artificial Intelligence
  (AAAI'08), AAAI Press, 2008, pp. 1062-1068.
• D. Poole, First-order probabilistic inference, in
  International Joint Conference On Artificial
  Intelligence, Lawrence Erlbaum Associates LTD,
  2003, pp. 985-991.
• P. Singla and P. Domingos, Lifted first-order
  belief propagation, in Proc. 23rd AAAI Conference
  on Artificial Intelligence (AAAI'08), AAAI Press,
  2008, pp. 1094-1099.

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THE END
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Queries
Full joint distribution specifies answer to any
query P(variable evidence about others)
Tuberculosis
Pneumonia
Lung Infiltrates
Sputum Smear
Sputum Smear
XRay
XRay