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Challenge Paper: Marginal Probabilities for Instances and Classes

This paper investigates the relationship between class-level and instance-level probabilistic queries within the context of artificial intelligence. It introduces the concept of marginal equivalence, illustrated through the example of bird flight probabilities. The analysis draws upon foundational works by Halpern and Bacchus, proposing that a system can achieve class-instance marginal equivalence by leveraging parametrized Bayes nets. The challenges associated with implementing this concept in statistical relational learning (SRL) systems are also discussed, emphasizing the significance of constraints to reduce computational complexity.

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Challenge Paper: Marginal Probabilities for Instances and Classes

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

  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. A connection between class-level and instance-level probabilities • 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%! Marginal Probabilities for Instances and Classes

  4. 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 1st-order variables and constants are typed:P(Flies(Tweety)) =P(Flies(B)). • The Marginal Equivalence Principle.

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

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

  7. II: Score Maximization Marginal Probabilities for Instances and Classesa

  8. III: Latent Variable Models Satisfy Marginal Equivalence Marginal Probabilities for Instances and Classesa U(S) U(C) intelligence(S) Registered(S,C) diff(C)

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

  10. 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. ParametrizedBayes Net Examples • 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. diff(C) intelligence(S) Registered(S,C) Marginal Probabilities for Instances and Classesa

  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. ParametrizedBayes Nets PBN + Marginal Equivalence Marginal Probabilities for Instances and Classesa

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