Cumulative Distribution Networks and
the Derivative-Sum-Product Algorithm
Jim C. Huang and Brendan J.
Frey
Probabilistic and Statistical Inference Group,
Department of Electrical and Computer Engineering,
University of Toronto,
Toronto, ON, Canada
UAI 2008
12/07/2008
Motivation
• Problems where density models may be intractable
• e.g.: Modelling arbitrary dependencies
e.g.: Predicting game outcomes in
Halo 2
• e.g.: Modelling stochastic orderings
•
Cumulative distribution network (CDN)
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12/07/2008
Cumulative distribution networks (CDNs)
• Graphical model of the cumulative distribution function (CDF)
• Example:
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12/07/2008
Cumulative distribution functions
Negative
convergence
Positive
convergence
Monotonicity
• Marginalization  maximization
• Conditioning  differentiation
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Necessary/sufficient
conditions on CDN functions
• Negative convergence (necessity and sufficiency):
For each Xk, at least one
neighboring function  0
• Positive convergence (sufficiency):
All functions  1
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12/07/2008
Necessary/sufficient conditions
on CDN functions
• Monotonicity lemma (sufficiency):
All functions monotonically
non-decreasing…
Sufficient condition for a valid joint CDF:
Each CDN function can be a CDF of its
arguments
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12/07/2008
Marginal independence
• Marginalization  maximization
– e.g.: X is marginally independent of Y
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Conditional independence
• Conditioning  differentiation
– e.g.: X and Y are conditionally dependent given Z
– e.g.: X and Y are conditionally independent given Z
• Conditional independence  No paths contain observed
variables
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12/07/2008
A toy example
Required “Bayes net”
Check:
Markov random fields
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Inference by message passing
• Conditioning  differentiation
• Replace sum in sum-product with differentiation
…
• Recursively apply product rule via message-passing with
messages , 
• Derivative-Sum-Product (DSP)
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Derivative-sum-product
• In a CDN:
• In a factor graph:
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Ranking in multiplayer gaming
• e.g.: Halo 2 game with 7 players, 3 teams
Player skill
functions
Player
performanc
e
Team
performanc
e
Given game outcomes, update player skills as a function of all player/team
performances
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12/07/2008
Ranking in multiplayer gaming
= Local cumulative model linking team rank rn
with player performances xn
e.g.: Team 2 has rank 2
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Ranking in multiplayer gaming
= Pairwise model of team ranks rn,rn+1
Enforce stochastic orderings between teams
via h
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Ranking in multiplayer gaming
• CDN functions = Gaussian CDFs
• Skill updates:
• Prediction:
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Results
• Previous methods for ranking players:
– ELO (Elo, 1978)
– TrueSkill (Graepel, Minka and Herbrich, 2006)
• After message-passing…
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12/07/2008
Summary
• The CDN as a graphical model for CDFs
• Unique conditional independence structure
• Marginalization  maximization
• Global normalization can be enforced locally
• Conditioning  differentiation
• Efficient inference with Derivative-Sum-Product
• Application to Halo 2 Beta Dataset
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12/07/2008
Discussion
• Need to be careful when applying to ordinal discrete
variables…
• Principled method for learning CDNs
• Variational principle? (loopy DSP seems to work well)
• Future applications to
– Hypothesis testing
– Document retrieval
– Collaborative filtering
– Biological sequence search
–…
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Thanks
• Questions?
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Interpretation of skill updates
• For any given player let
denote the
outcomes of games he/she has played previously
• Then the skill function corresponds to
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12/07/2008