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Logical Bayesian Networks

Logical Bayesian Networks. A knowledge representation view on Probabilistic Logical Models. Daan Fierens, Hendrik Blockeel, Jan Ramon, Maurice Bruynooghe Katholieke Universiteit Leuven, Belgium. THIS TALK. learning best known most developed. Probabilistic Logical Models.

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Logical Bayesian Networks

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  1. Logical Bayesian Networks A knowledge representation view on Probabilistic Logical Models Daan Fierens, Hendrik Blockeel, Jan Ramon, Maurice Bruynooghe Katholieke Universiteit Leuven, Belgium

  2. THIS TALK • learning • best known • most developed Probabilistic Logical Models • Variety of PLMs: • Origin in Bayesian Networks (Knowledge Based Model Construction) • Probabilistic Relational Models • Bayesian Logic Programs • CLP(BN) • … • Origin in Logic Programming • PRISM • Stochastic Logic Programs • …

  3. Combining PRMs and BLPs • PRMs: • + Easy to understand, intuitive • - Somewhat restricted (as compared to BLPs) • BLPs: • + More general, expressive • - Not always intuitive • Combine strengths of both models in one model ? • We propose Logical Bayesian Networks (PRMs+BLPs)

  4. Overview of this Talk • Example • Probabilistic Relational Models • Bayesian Logic Programs • Combining PRMs and BLPs: Why and How ? • Logical Bayesian Networks

  5. Example [ Koller et al.] • University: • students (IQ) + courses (rating) • students take courses (grade) • grade  IQ • rating  sum of IQ’s • Specific situation: • jeff takes ai, pete and rick take lp, no student takes db

  6. rating(db) rating(ai) rating(lp) iq(jeff) iq(pete) iq(rick) grade(jeff,ai) grade(rick,lp) grade(pete,lp) Bayesian Network-structure

  7. Course Student iq rating Takes grade PRMs [Koller et al.] • PRM: relational schema,dependency structure (+ aggregates + CPDs) CPT aggr + CPT

  8. PRMs (2) • Semantics: PRM induces a Bayesian network on the relational skeleton

  9. rating(db) rating(ai) rating(lp) iq(jeff) iq(pete) iq(rick) grade(jeff,ai) grade(rick,lp) grade(pete,lp) PRMs - BN-structure (3)

  10. PRMs: Pros & Cons (4) • Easy to understand and interpret • Expressiveness as compared to BLPs, … : • Not possible to combine selection and aggregation [Blockeel & Bruynooghe, SRL-workshop ‘03] • E.g. extra attribute sex for students • rating  sum of IQ’s for female students • Specification of logical background knowledge ? • (no functors, constants)

  11. BLPs [Kersting, De Raedt] • Definite Logic Programs + Bayesian networks • Bayesian predicates (range) • Random var = ground Bayesian atom: iq(jeff) • BLP = clauses with CPT rating(C) | iq(S), takes(S,C). CPT + combining rule (can be anything) Range: {low,high} • Semantics: Bayesian network • random variables = ground atoms in LH-model • dependencies  grounding of the BLP

  12. BLPs (2) rating(C) | iq(S), takes(S,C). rating(C) | course(C). grade(S,C) | iq(S), takes(S,C). iq(S) | student(S). • BLPs do not distinguish probabilistic and logical/certain/structural knowledge • Influence on readability of clauses • What about the resulting Bayesian network ? student(pete)., …, course(lp)., …, takes(rick,lp).

  13. takes(jeff,ai) student(jeff) CPD iq(jeff) grade(jeff,ai) student(jeff) iq(jeff) true distribution for iq/1 false ? BLPs - BN-structure (3) • Fragment:

  14. takes(jeff,ai) student(jeff) iq(jeff) takes(jeff,ai) grade(jeff,ai) grade(jeff,ai) true distribution for grade/2, function of iq(jeff) CPD false ? BLPs - BN-structure (3) • Fragment:

  15. BLPs: Pros & Cons (4) • High expressiveness: • Definite Logic Programs (functors, …) • Can combine selection and aggregation (combining rules) • Not always easy to interpret • the clauses • the resulting Bayesian network

  16. Combining PRMs and BLPs • Why ? • 1 model = intuitive + high expressiveness • How ? • Expressiveness: ( BLPs) • Logic Programming • Intuitive: ( PRMs) • Distinguish probabilistic and logical/certain knowledge • Distinct components (PRMs: schema determines random variables / dependency structure) • (General vs Specific knowledge)

  17. Logical Bayesian Networks • Probabilistic predicates (variables,range) vs Logical predicates • LBN - components: • Relational schema  V • Dependency Structure  DE • CPDs+ aggregates  DI • Relational skeleton  Logic Program Pl • Description of DoD / deterministic info

  18. Logical Bayesian Networks • Semantics: • LBN induces a Bayesian network on the variables determined by Pl and V

  19. Normal Logic Program Pl student(jeff). course(ai). takes(jeff,ai). student(pete). course(lp). takes(pete,lp). student(rick). course(db). takes(rick,lp). • Semantics: well-founded model WFM(Pl)(when no negation: least Herbrand model)

  20. V iq(S) <= student(S). rating(C) <= course(C). grade(S,C) <= takes(S,C). • Semantics: determines random variables • each ground probabilistic atom in WFM(Pl V) is random variable • iq(jeff), …, rating(lp), …,grade(rick,lp) • non-monotonic negation (not in PRMs, BLPs) • grade(S,C) <= takes(S,C), not(absent(S,C)).

  21. DE grade(S,C) | iq(S). rating(C) | iq(S) <- takes(S,C). • Semantics: determines conditional dependencies • ground instances with context in WFM(Pl) • e.g. rating(lp) | iq(pete) <- takes(pete,lp) • e.g. rating(lp) | iq(rick) <- takes(rick,lp)

  22. V + DE iq(S) <= student(S). rating(C) <= course(C). grade(S,C) <= takes(S,C) grade(S,C) | iq(S). rating(C) | iq(S) <- takes(S,C).

  23. rating(db) rating(ai) rating(lp) iq(jeff) iq(pete) iq(rick) grade(jeff,ai) grade(rick,lp) grade(pete,lp) LBNs - BN-structure

  24. DI • The quantitative component • ~ in PRMs: aggregates + CPDs • ~ in BLPs: CPDs + combining rules • For each probabilistic predicate p a logical CPD • = function with • input: set of pairs (Ground prob atom,Value) • output: probability distribution for p • Semantics: determines the CPDs for all variables about p

  25. 0.5 / 0.5 0.7 / 0.3 DI (2) • e.g. for rating/1 (inputs are about iq/1) If (SUM(iq(S),Val) Val) > 1000 Then 0.7 high / 0.3 low Else 0.5 high / 0.5 low • Can be written as logical probability tree (TILDE) sum(Val, iq(S,Val), Sum), Sum > 1000 • cf [Van Assche et al., SRL-workshop ‘04]

  26. DI (3) • DI determines the CPDs • e.g. CPD for rating(lp) = function of iq(pete) and iq(rick) • Entry in CPD for iq(pete)=100 and iq(rick)=120 ? • Apply logical CPD for rating/1 to {(iq(pete),100),(iq(rick),120)} • Result: probab. distribution 0.5 high / 0.5 low If (SUM(iq(S),Val) Val) > 1000 Then 0.7 high / 0.3 low Else 0.5 high / 0.5 low

  27. 0.5 / 0.5 0.7 / 0.3 DI (4) • Combine selection and aggregation? • e.g. rating  sum of IQ’s for female students sum(Val, (iq(S,Val), sex(S,fem)), Sum), Sum > 1000 • again cf [Van Assche et al., SRL-workshop ‘04]

  28. LBNs: Pros & Cons / Conclusion • Qualitative part (V + DE): easy to interpret • High expressiveness • Normal Logic Programs (non-monotonic negation, functors, …) • Combining selection and aggregation • Comes at a cost: • Quantitative part (DI) is more difficult (than for PRMs)

  29. Future Work: Learning LBNs • Learning algorithms for PRMs & BLPs • On high level: appropriate mix will probably do for LBNs • LBNs  PRMs: learning quantitative component is more difficult for LBNs • LBNs  BLPs: • LBNs have separation V vs DE • LBNs: distinction probabilistic predicates vs logical predicates = bias (but also used by BLPs in practice)

  30. ?

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