12/07/2008UAI 2008

Cumulative Distribution Networks and the Derivative-

Sum-Product Algorithm

Jim C. Huang and Brendan J. FreyProbabilistic and Statistical Inference Group, Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON, Canada

12/07/2008UAI 2008

• Problems where density models may be intractable

• e.g.: Modelling arbitrary dependencies

• e.g.: Modelling stochastic orderings

• Cumulative distribution network (CDN)

Motivation

e.g.: Predicting game outcomes in Halo 2

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Cumulative distribution networks (CDNs)

• Graphical model of the cumulative distribution function (CDF)

• Example:

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Positive convergence

Negative convergence

Monotonicity

Cumulative distribution functions

• Marginalization maximization

• Conditioning differentiation

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Necessary/sufficient conditions on CDN

functions• Negative convergence (necessity and sufficiency):

• Positive convergence (sufficiency):

For each Xk, at least one neighboring function 0

All functions 1

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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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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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Check:

A toy example

Markov random fields

Required “Bayes net”

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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 performance

Team performance

Given game outcomes, update player skills as a function of all player/team performances

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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

Enforce stochastic orderings between teams via h

= Pairwise model of team ranks rn,rn+1

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• CDN functions = Gaussian CDFs

• Skill updates:

• Prediction:

Ranking in multiplayer gaming

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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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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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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