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Modelling Relational Statistics With Bayes Nets

School of Computing Science Simon Fraser University Vancouver, Canada. Modelling Relational Statistics With Bayes Nets. Class-Level and Instance-Level Queries. Classic AI research distinguished two types of probabilistic relational queries. ( Halpern 1990, Bacchus 1990). .

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Modelling Relational Statistics With Bayes Nets

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  1. School of Computing Science • Simon Fraser University • Vancouver, Canada Modelling Relational Statistics With Bayes Nets

  2. Class-Level and Instance-Level Queries • Classic AI research distinguished two types of probabilistic relational queries. (Halpern 1990, Bacchus 1990). Class-level queries Relational Statistics Type 1 probabilities Instance-level queries Ground facts Type 2 probabilities Relational Query Halpern, “An analysis of first-order logics of probability”, AI Journal 1990.Bacchus, “Representing and reasoning with probabilistic knowledge”, MIT Press 1990.

  3. Visualizing Class-Level Probability • Percentage of Flying Birds = 90%. • Halpern: Probability that a typical or random bird flies is 90%. Syntactic Distinction • Contains some free variables. • e.g. P(Flies(B)) = ?. • Contains no free variables. • e.g. P(Flies(tweety)) = ?. Modelling Relational Statistics With Bayes Nets

  4. Applications of Class-Level Modelling • 1st-order rule learning (e.g., “intelligent students take difficult courses”). • Strategic Planning (e.g., “increase SAT requirements to decrease student attrition”). • Query Optimization (Getoor, Taskar, Koller 2001). Class-level queries support selectivity estimation  optimal evaluation order for SQL query . Getoor, Lise, Taskar, Benjamin, and Koller, Daphne. Selectivity estimation using probabilistic models. ACM SIGMOD Record, 30(2):461–472, 2001.

  5. No Grounding Semantics for Class-level Queries • “Unrolling” a network → model of individual entities. • No classes, cannot ask class-level queries. Registered(jack,100) intelligence(jack) intelligence(S) intelligence(jane) Registered(jack,200) Registered(S,C) diff(100) diff(C) Registered(jane,100) diff(200) Registered(jane,200) Instance-level Model w/domain(S) = {jack,jane}domain(C) = {100,200} Class-level Templatewith 1st-order Variables Modelling Relational Statistics With Bayes Netsa

  6. Previous Work: Probabilistic Queries in Statistical-Relational Learning Class-Level Instance-Level Statistical-Relational Models (LiseGetoor, Taskar, Koller 2001) Many Model Types: Probabilistic Relational Models, Markov Logic Networks, Bayes Logic Programs, Logical Bayesian Networks, …

  7. New Unified Approach Class-Level Instance-Level Parametrized Bayes Nets + new class-level semantics Parametrized Bayes Nets + combining rules (Poole 2003) + log-linear model (Khosravi, Schulte et al. 2010, Schulte and Khosravi 2012) David Poole, “First-Order Probabilistic Inference”, IJCAI 2003. H. Khosravi, O. Schulte, T. Man, X. Xu, and B. Bina, “Structure learning for Markov logic networks with many descriptive attributes”, in AAAI, 2010. O. Schulte and H. Khosravi. “Learning graphical models for relational data via lattice search”. Machine Learning, 2012.

  8. Random Selection Semantics: Example • Apply the random selection semantics for probabilistic 1st-order logic (Halpern 1990; Bacchus 1990). intelligence(S) diff(C) hi hi Registered(S,C) true P(intelligence(S) = hi, diff(C) = hi, Registered(S,C) = true) = 20% means: “if we randomly select a student and a course, then the probability is 20% that the student is registered in the course, and that the intelligence of the student and the difficulty of the course are high.” Halpern, “An analysis of first-order logics of probability”, AI Journal 1990.Bacchus, “Representing and reasoning with probabilistic knowledge”, MIT Press 1990.

  9. Computing Parameter Estimates (I) • Use conditional database probabilities as Bayes net parameters. • Maximizes the random selection pseudo-likelihood (Schulte 2011). • For database probabilities with all true relationships, use SQL or Virtual Join (Yin, Han et al. 2004). R1 R2 Schulte, O. “A tractable pseudo-likelihood function for Bayes nets applied to relational data.” SIAM SDM, 2011. Yin, X., Han. J. et al. “CrossMine: Efficient Classification Across Multiple Database Relations”.Constraint-Based Mining and Inductive Databases, 2004.

  10. Computing Parameter Estimates (II) • How to compute database probabilities for negated relations? • e.g., number of U.S. users who are not friends? • Materializing complement tables is unscalable. • For single false relation, “1-minus trick” (Getoor et al. 2007). • General case: New application of the fast Möbius transform (Kennes and Smits 1990). Getoor, Lise, Friedman, Nir, Koller, Daphne, Pfeffer, Avi, and Taskar, Benjamin. Probabilistic relational models, 2007. Kennes, Robert and Smets, Philippe. Computational aspects of the Mobius transformation. In UAI, 1990.

  11. The MöbiusParametrization For two link types Joint probabilities Count(*) Count(*) Count(*) Count(*) Count(*) R1 Count(*) R1 R1 R1 R1 R1 R2 R2 R2 R2 R2 Möbius Parameters R2 Count(*) Count(*) no condition Modelling Relational Statistics With Bayes Netsa

  12. Evaluation • Fast: parameters in minutes or less. • Accurate queries/estimates. • Try it yourself in our demo! Modelling Relational Statistics With Bayes Nets

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