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First-Order Probabilistic Inference

This presentation by Rodrigo de Salvo Braz from SRI International discusses the goal of first-order probabilistic inference, propositionalization, lifted inference, inversion elimination, counting elimination, DBLOG, BLOG background, retro-instantiation, and improving BLOG inference. It also explores future directions and conclusions.

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First-Order Probabilistic Inference

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  1. First-OrderProbabilistic Inference Rodrigo de Salvo Braz SRI International

  2. A remark Feel free to ask clarification questions!

  3. Slides online • Just search for“Rodrigo de Salvo Braz” and check at the presentations page. (www.cs.berkeley.edu/~braz)

  4. Outline • A goal: First-order Probabilistic Inference • An ultimate goal • Propositionalization • Lifted Inference • Inversion Elimination • Counting Elimination • DBLOG • BLOG background • DBLOG • Retro-instantiation • Improving BLOG inference • Future Directions and Conclusion

  5. Outline • A goal: First-order Probabilistic Inference • An ultimate goal • Propositionalization • Lifted Inference • Inversion Elimination • Counting Elimination • DBLOG • BLOG background • DBLOG • Retro-instantiation • Improving BLOG inference • Future Directions and Conclusion

  6. Abstracting Inference and Learning AI Problems ... Commonsense reasoning Natural LanguageProcessing Planning Vision How to solvecommonsensereasoning DomainKnowledge How to solveNaturalLanguageProcessing LanguageKnowledge Domain &PlanningKnowledge How to solvePlanning Objects &OpticsKnowledge How to solveVision InferenceandLearning InferenceandLearning InferenceandLearning InferenceandLearning InferenceandLearning

  7. Inference and Learning InferenceandLearning What capabilities should it have? • It should represent and use: • predicates (relations) and quantification over objects: • 8 X male(X) Ç female(X) • 8 X male(X) ) sick(X) • sick(p121) • varying degrees of uncertainty: • 8 X { male(X) ) sick(X) } • since it may hold often but not absolutely. • essential for language, vision, pattern recognition, commonsense reasoning etc • modal knowledge, utilities, and more.

  8. Abstracting Inference and Learning Commonsense reasoning Natural Language Processing • Domain knowledge: • male(X) Ç female(X) • { male(X) ) sick(X) } • sick(o121) • ... Language knowledge: { sentence(S) Æverb(S,”had”) Æ object(S,”fever”) subject(S,X) ) fever(X) } ... Sharing inference and learning module InferenceandLearning

  9. Abstracting Inference and Learning Commonsense reasoning WITH Natural Language Processing • Domain knowledge: • male(X) Ç female(X) • { male(X) ) sick(X) } • sick(o121) • ... Language knowledge: { sentence(S) Æverb(S,”had”) Æ object(S,”fever”) subject(S,X) ) fever(X) } verb(s13,”had”), object(S, “fever”), subject(s13,o121)... Joint solution made simpler InferenceandLearning

  10. Standard solutions fall short • Logic • Has objects and properties; • but statements are absolute; • Graphical models and machine learning • Have varying degrees of uncertainty; • but propositional; treating objects and quantification is awkward.

  11. Standard solutions fall short Graphical models and objects sick_o135 • male(X) Ç female(X) • { male(X) )sick(X) } • { friend(X,Y) Æ male(X) ) male(Y) } male_o135 female_o135 • sick(o121) sick(o135) friend_o121_o135 • Transformation (propositionalization) not part of graphical model framework; • Graphical model have only random variables, not objects; • Different data creates distinct, formally unrelated model. male_o121 female_o121 sick_121

  12. Outline • A goal: First-order Probabilistic Inference • An ultimate goal • Propositionalization • Lifted Inference • Inversion Elimination • Counting Elimination • DBLOG • BLOG background • DBLOG • Retro-instantiation • Improving BLOG inference • Future Directions and Conclusion

  13. First-Order Probabilistic Inference • Many languages have been proposed • Knowledge-based model construction (Breese, 1992) • Probabilistic Logic Programming (Ng & Subrahmanian, 1992) • Probabilistic Logic Programming (Ngo and Haddawy, 1995) • Probabilistic Relational Models (Friedman et al., 1999) • Relational Dependency Networks (Neville & Jensen, 2004) • Bayesian Logic (BLOG) (Milch & Russell, 2004) • Bayesian Logic Programs (Kersting & DeRaedt, 2001) • DAPER (Heckerman, Meek & Koller, 2007) • Markov Logic Networks (Richardson & Domingos, 2004) • “Smart” propositionalization is the main method.

  14. Propositionalization • Bayesian Logic Programs Prolog-like statements such as medicine(Hospital) | in(Patient,Hospital), sick(Patient). sick(Patient) | in(Patient,Hospital), exposed(Hospital).associated with CPTs and combination rules. |instead of :-

  15. Propositionalization medicine(Hospital) | in(Patient,Hospital), sick(Patient). (plus CPT) BN grounded from and-or tree: in(john,hosp13) ... ... sick(john) medicine(hosp13) in(peter,hosp13) ... sick(peter) ... ...

  16. Propositionalization Multi-Entity Bayesian Networks (Laskey, 2004) in(Patient,Hospital) exposed(Hospital) sick(Patient) in(Patient,Hospital) sick(Patient) medicine(Hospital)

  17. in(Patient,Hospital) in(john,hosp13) in(peter,hosp13) sick(Patient) sick(john) sick(peter) medicine(Hospital) medicine(hosp13) medicine(hosp13) Propositionalization Multi-Entity Bayesian Networks (Laskey, 2004) first-order fragments instantiated ...

  18. in(john,hosp13) in(peter,hosp13) sick(john) sick(peter) in(john,hosp13) sick(john) in(peter,hosp13) sick(peter) medicine(hosp13) medicine(hosp13) medicine(hosp13) Propositionalization Multi-Entity Bayesian Networks (Laskey, 2004) ... ...

  19. Outline • A goal: First-order Probabilistic Inference • An ultimate goal • Propositionalization • Lifted Inference • Inversion Elimination • Counting Elimination • DBLOG • BLOG background • DBLOG • Retro-instantiation • Improving BLOG inference • Future Directions and Conclusion

  20. Lifted Inference medicine(Hospital) ( in(Patient,Hospital) Æ sick(Patient) { sick(Patient) ( in(Patient,Hospital) Æ exposed(Hospital) } • Faster • More compact • More intuitive • Higher level – more information/structure available for optimization Unbound, general variable in(Patient, hosp13) medicine(hosp13) exposed(hosp13) sick(Patient)

  21. Bayesian Networks (directed) P(epidemic) epidemic P(sick_john | epidemic) P(sick_bob | epidemic) sick_john sick_mary sick_bob P(sick_mary | epidemic) P(sick_john, sick_mary, sick_bob, epidemic)= P(sick_john | epidemic) * P(sick_mary | epidemic) * P(sick_bob | epidemic) * P(epidemic)

  22. Factor Networks (undirected) 2 epidemic_france epidemic_uk 3 factor,or potential function 1 epidemic_belgium epidemic_germany 4 P(epi_france, epi_belgium, epi_uk, epi_germany)/1(epi_france,epi_belgium) * 2(epi_france, epi_uk) * 3(epi_france,epi_germany) * 4(epi_belgium,epi_germany)

  23. Bayesian Nets as Factor Networks P(epidemic) epidemic P(sick_john | epidemic) P(sick_bob | epidemic) sick_john sick_mary sick_bob P(sick_mary | epidemic) P(sick_john, sick_mary, sick_bob, epidemic)/ P(sick_john | epidemic) * P(sick_mary | epidemic) * P(sick_bob | epidemic) * P(epidemic)

  24. Inference: Marginalization P(epidemic) epidemic P(sick_john | epidemic) P(sick_bob | epidemic) sick_john sick_mary sick_bob P(sick_mary | epidemic) P(sick_john) /epidemicsick_marysick_bob P(sick_john | epidemic) * P(sick_mary | epidemic) * P(sick_bob | epidemic) * P(epidemic)

  25. Inference: Variable Elimination (VE) P(epidemic) epidemic P(sick_john | epidemic) P(sick_bob | epidemic) sick_john sick_mary sick_bob P(sick_mary | epidemic) P(sick_john) /epidemic P(sick_john | epidemic) * P(epidemic) * sick_maryP(sick_mary | epidemic) * sick_bob P(sick_bob | epidemic)

  26. Inference: Variable Elimination (VE) P(epidemic) epidemic P(sick_john | epidemic) 1(epidemic) sick_john sick_mary P(sick_mary | epidemic) P(sick_john) /epidemic P(sick_john | epidemic) * P(epidemic) * sick_maryP(sick_mary | epidemic) * 1 (epidemic)

  27. Inference: Variable Elimination (VE) P(epidemic) epidemic P(sick_john | epidemic) 1(epidemic) sick_john sick_mary P(sick_mary | epidemic) P(sick_john) /epidemic P(sick_john | epidemic) * P(epidemic) * 1 (epidemic) * sick_maryP(sick_mary | epidemic)

  28. Inference: Variable Elimination (VE) P(epidemic) epidemic P(sick_john | epidemic) 1(epidemic) sick_john 2(epidemic) P(sick_john) /epidemic P(sick_john | epidemic) * P(epidemic) * 1 (epidemic) * 2 (epidemic)

  29. Inference: Variable Elimination (VE) 3(sick_john) sick_john P(sick_john) /3 (sick_john)

  30. A Factor Network epidemic(measles) … epidemic(flu) … … sick(mary,measles) sick(mary,flu) sick(bob,measles) sick(bob,flu) … … … … … hospital(mary) hospital(bob) …

  31. First-order Representation Logical Variables parameterize random variables epidemic(D) Parfactors, for parameterized factors sick(P,D) Atoms represent a set of random variables P  john Contraints to logical variables hospital(P)

  32. Semantics epidemic(measles) … epidemic(flu) … … sick(mary,measles) sick(mary,flu) sick(bob,measles) sick(bob,flu) … … … f(sick(mary,measles), epidemic(measles)) … … hospital(mary) hospital(bob) …

  33. Outline • A goal: First-order Probabilistic Inference • An ultimate goal • Propositionalization • Lifted Inference • Inversion Elimination • Counting Elimination • DBLOG • BLOG background • DBLOG • Retro-instantiation • Improving BLOG inference • Future Directions and Conclusion

  34. Inversion Elimination (IE) D  measles D  measles epidemic(D) epidemic(D) IE D  measles D  measles sick(P, D) sick(P, D) Abstraction Grounding sick(P,D)(epidemic(D),sick(P,D)) epidemic(flu) epidemic(rubella) epidemic(flu) epidemic(rubella) … … VE sick(john, flu) sick(mary, rubella) sick(john, flu) sick(mary, rubella) … …

  35. Inversion Elimination (IE) • Proposed by Poole (2003); • Lacked formalization; • Did not identify cases in which it is incorrect; • Formalized and identified correctness conditions (with Dan Roth and Eyal Amir).

  36. Inversion Elimination - Limitations • Requires eliminated RVs to occur in separate instances of parfactor epidemic(D) InversionElimination correct D  measles sick(mary, D) … epidemic(flu) epidemic(rubella) … sick(mary, flu) sick(mary, rubella)

  37. epidemic(D1) month D1  D2 epidemic(D2) Inversion Elimination - Limitations • Requires eliminated RVs to occur in separate instances of parfactor IE not correct epidemic(D1) month D1  D2 epidemic(D2) month epidemic(flu) … epidemic(measles) epidemic(rubella)

  38. Outline • A goal: First-order Probabilistic Inference • An ultimate goal • Propositionalization • Lifted Inference • Inversion Elimination • Counting Elimination • DBLOG • BLOG background • DBLOG • Retro-instantiation • Improving BLOG inference • Future Directions and Conclusion

  39. epidemic(D1) month D1  D2 month epidemic(D2) epidemic(flu) … epidemic(measles) epidemic(rubella) Counting Elimination • Need to consider joint assignments; • Exponential number of those; • But actually, potential depends on histogram of values in assignment only: 00101 the same as 11000; • Polynomial number of assignments instead.

  40. A Simple Experiment

  41. Outline • A goal: First-order Probabilistic Inference • An ultimate goal • Propositionalization • Lifted Inference • Inversion Elimination • Counting Elimination • DBLOG • BLOG background • DBLOG • Retro-instantiation • Improving BLOG inference • Future Directions and Conclusion

  42. Outline • A goal: First-order Probabilistic Inference • An ultimate goal • Propositionalization • Lifted Inference • Inversion Elimination • Counting Elimination • DBLOG • BLOG background • DBLOG • Retro-instantiation • Improving BLOG inference • Future Directions and Conclusion

  43. BLOG (Bayesian LOGic) • Milch & Russell (2004); • A probabilistic logic language; • Current inference is propositional sampling; • Special characteristics: • Open Universe; • Expressive language.

  44. BLOG (Bayesian LOGic) type Hospital; #Hospital ~ Uniform(1,10); random Boolean Large(Hospital hosp) ~ Bernoulli(0.6); random NaturalNum Region(Hospital hosp) ~ TabularCPD[[0.3, 0.4, 0.2, 0.1]](); type Patient; #Patient(HospitalOf = Hospital hosp)if Large(hosp) then ~ Poisson(1500); else if Region(hosp) = 2 then = 300; else ~ Poisson(500); query Average( {#Patient(b) for Hospital b} ); Open Universe Expressive language

  45. Inference in BLOG random Boolean Exposed(hosp) ~ Bernoulli(0.7); random Boolean Sick(Patient patient)if Exposed(HospitalOf(patient)) then if Male(patient) then ~ Bernoulli(0.1); else ~ Bernoulli(0.4); else = False; random Boolean Male(Patient patient) ~ Bernoulli(0.5); random Boolean Medicine(Hospital hosp)= exists Patient patient HospitalOf(patient) = hosp & Sick(patient); guaranteed Hospital hosp13; query Medicine(hosp13);

  46. Inference in BLOG - Sampling Open Universe Exposed(hosp13) #Patient(hosp13) Sampled false Sick(patient1) Medicine(hosp13) false ... false Sick(patient73) false

  47. Inference in BLOG - Sampling Open Universe Exposed(hosp13) #Patient(hosp13) Sampled true Sick(patient1) Male(patient1) Medicine(hosp13) true true true

  48. Outline • A goal: First-order Probabilistic Inference • An ultimate goal • Propositionalization • Lifted Inference • Inversion Elimination • Counting Elimination • DBLOG • BLOG background • DBLOG • Retro-instantiation • Improving BLOG inference • Future Directions and Conclusion

  49. Temporal models in BLOG • Expressive enough to directly write temporal models type Aircraft; #Aircraft ~ Poisson[3]; random Real Position(Aircraft a, NaturalNum t) if t = 0 then ~ UniformReal[-10, 10]() else ~ Gaussian(Position(a, Pred(t)), 2); type Blip; // num of blips from aircraft a is 0 or 1 #Blip(Source = Aircraft a, Time = NaturalNum t) ~ TabularCPD[[0.2, 0.8]]() // num false alarms has Poisson distribution. #Blip(Time = NaturalNum t) ~ Poisson[2]; random Real BlipPosition(Blip b, NaturalNum t) ~ Gaussian(Position(Source(b), Pred(t)), 2));

  50. Temporal models in BLOG • Inference algorithm does not use temporal structure: • Markov property: state depends on previous state only; • evidence and query come for successive time steps; • Dynamic BLOG (DBLOG); • analogous to Dynamic Bayesian networks (DBNs).

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