1. Challenge Paper: Marginal Probabilities for Instances and Classes Oliver Schulte School of Computing Science Simon Fraser University Vancouver, Canada

2. 2/12 Class-Level and Instance-Level Queries Classic AI research distinguished two types of probabilistic relational queries. (Halpern 1990, Bacchus 1990). Halpern, “An analysis of first-order logics of probability”, AI Journal 1990. Bacchus, “Representing and reasoning with probabilistic knowledge”, MIT Press 1990. Relational Query Class-level QueryReference Class What is the percentage of flying birds? Birds What is the percentage of friendship pairs where both are women? Pairs of Friends What is the percentage of A grades awarded to highly intelligence students? Student-course pairs where student is registered in course. Instance-Level Query Given that Tweety is a bird, what is the probability that Tweety flies? Given that Sam and Hilary are friends, and given the genders of their other friends, what is the probability that Sam and Hilary are both women? What is the probabiity that Jack is highly intelligent given his grades? Instance-level queries Ground facts Type 2 probabilities Class-level queries Relational Statistics Type 1 probabilities

3. 3/12 A connection between class-level and instance-level probabilities Marginal Probabilities for Instances and Classes Percentage of Flying Birds = 90%. Halpern: Probability that a typical or random bird flies is 90%. What is the answer to P(Flies(Tweety))? It should be 90%!

4. 4/12 Halpern’s Instantiation Version Given that Tweety is a bird (and nothing else), the probability that Tweety flies = the probability that a randomly chosen bird flies. P(Flies(Tweety)|Bird(Tweety)) = P(Flies(B)|Bird(B)). Assuming that 1 st -order variables and constants are typed: P(Flies(Tweety)) =P(Flies(B)). The Marginal Equivalence Principle.

5. 5/12 Marginal Probabilities for Instances and Classesa Four Arguments for Marginal Equivalence

6. 6/12 I: Intuitive Plausibility Used in cold-start problems. Equivalent to Miller’s principle. Marginal Probabilities for Instances and Classesa

7. 7/12 II: Score Maximization Marginal Probabilities for Instances and Classesa CourseID difficultyP 1 (diff)P 2 (diff) 100lo1/31/2 200hi2/31/2 300hi2/31/2 400lo1/31/2 Score4/27≈ 0.15 1/16≈ 0.17

8. 8/12 III: Latent Variable Models Satisfy Marginal Equivalence Marginal Probabilities for Instances and Classesa intelligence(S)diff(C)Registered(S,C) U(S)U(C) CourseID difficultyU(C) 100lo10 200hi20 300hi15 400lo12 SNameint.U(S) Annalo30 Bobhi20

9. 9/12 IV: Something Else Marginal Probabilities for Instances and Classesa

10. 10/12 The Challenge If we accept that an SRL system should satisfy class- instance marginal equivalence, how do we design a system to achieve that? Marginal Probabilities for Instances and Classesa

11. 11/12 Parametrized Bayes Net Examples Marginal Probabilities for Instances and Classesa intelligence(S) diff(C) Registered(S,C) Proposition If each node in the ground network has a unique set of parents, then class-level marginals = instance-level marginal. For other structures, it depends on the combining rule/parameters used.

12. 12/12 Constraints are Good Pedro Domingos: “The search space for SRL algorithms is very large even by AI standards.”  Class-instance marginal equivalence reduces the search space. Strong theoretical foundation. The challenge is to implement the constraint. Marginal Probabilities for Instances and Classesa Parametrized Bayes Nets PBN + Marginal Equivalence