Challenge Paper: Marginal Probabilities
for Instances and Classes
Oliver Schulte
School of Computing Science
Simon Fraser University
Vancouver, Canada
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
Relational Query
Class-level Query
Reference 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 queries
Ground facts
Type 2 probabilities
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?
Halpern, “An analysis of first-order logics of probability”, AI Journal 1990.
Bacchus, “Representing and reasoning with probabilistic knowledge”, MIT Press 1990.
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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
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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.
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Four Arguments for Marginal Equivalence
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I: Intuitive Plausibility
 Used in cold-start problems.
 Equivalent to Miller’s principle.
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II: Score Maximization
CourseID
difficulty
P1(diff)
P2(diff)
100
lo
1/3
1/2
200
hi
2/3
1/2
300
hi
2/3
1/2
400
lo
1/3
1/2
Score
4/27≈
0.15
1/16≈
0.17
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III: Latent Variable Models Satisfy
Marginal Equivalence
U(S)
U(C)
intelligence(S)
Registered(S,C)
CourseID
difficulty
U(C)
100
lo
10
200
hi
300
400
diff(C)
SName int.
U(S)
20
Anna
lo
30
hi
15
Bob
hi
20
lo
12
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IV: Something Else
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The Challenge
If we accept that an SRL system should satisfy classinstance marginal equivalence, how do we design
a system to achieve that?
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Parametrized Bayes Net Examples
diff(C)
intelligence(S)
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.
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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.
Parametrized Bayes Nets
PBN + Marginal Equivalence
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