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Cumulative Distribution Networks and the Derivative-Sum-Product Algorithm. Jim C. Huang and Brendan J. Frey

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Cumulative distribution networks and the derivative sum product algorithm

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


Motivation
Motivation Derivative-Sum-Product Algorithm

e.g.: Predicting game outcomes in Halo 2

  • Problems where density models may be intractable

  • e.g.: Modelling arbitrary dependencies

  • e.g.: Modelling stochastic orderings

  • Cumulative distribution network (CDN)

UAI 2008


Cumulative distribution networks cdns
Cumulative distribution networks (CDNs) Derivative-Sum-Product Algorithm

  • Graphical model of the cumulative distribution function (CDF)

  • Example:

UAI 2008


Cumulative distribution functions
Cumulative distribution functions Derivative-Sum-Product Algorithm

Negative convergence

  • Marginalization  maximization

  • Conditioning  differentiation

Positive convergence

Monotonicity

UAI 2008


Necessary sufficient conditions on cdn functions
Necessary/sufficient conditions on CDN functions Derivative-Sum-Product Algorithm

  • Negative convergence (necessity and sufficiency):

  • Positive convergence (sufficiency):

For each Xk, at least one neighboring function  0

All functions  1

UAI 2008


Necessary sufficient conditions on cdn functions1
Necessary/sufficient conditions on CDN functions Derivative-Sum-Product Algorithm

  • 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

UAI 2008


Marginal independence
Marginal independence Derivative-Sum-Product Algorithm

  • Marginalization  maximization

    • e.g.: X is marginally independent of Y

UAI 2008


Conditional independence
Conditional independence Derivative-Sum-Product Algorithm

  • 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

UAI 2008


A toy example
A toy example Derivative-Sum-Product Algorithm

Required “Bayes net”

Markov random fields

Check:

UAI 2008


Inference by message passing
Inference by message passing Derivative-Sum-Product Algorithm

  • Conditioning  differentiation

  • Replace sum in sum-product with differentiation

  • Recursively apply product rule via message-passing with messages ,

  • Derivative-Sum-Product (DSP)

UAI 2008


Derivative sum product
Derivative-sum-product Derivative-Sum-Product Algorithm

  • In a CDN:

  • In a factor graph:

UAI 2008


Ranking in multiplayer gaming
Ranking in multiplayer gaming Derivative-Sum-Product Algorithm

Player skill functions

Player performance

Team performance

  • e.g.: Halo 2 game with 7 players, 3 teams

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

UAI 2008


Ranking in multiplayer gaming1
Ranking in multiplayer gaming Derivative-Sum-Product Algorithm

= Local cumulative model linking team rank rn

with player performances xn

e.g.: Team 2 has rank 2

UAI 2008


Ranking in multiplayer gaming2
Ranking in multiplayer gaming Derivative-Sum-Product Algorithm

= Pairwise model of team ranks rn,rn+1

Enforce stochastic orderings between teams via h

UAI 2008


Ranking in multiplayer gaming3
Ranking in multiplayer gaming Derivative-Sum-Product Algorithm

  • CDN functions = Gaussian CDFs

  • Skill updates:

  • Prediction:

UAI 2008


Results
Results Derivative-Sum-Product Algorithm

  • Previous methods for ranking players:

    • ELO (Elo, 1978)

    • TrueSkill (Graepel, Minka and Herbrich, 2006)

  • After message-passing…

UAI 2008


Summary
Summary Derivative-Sum-Product Algorithm

  • 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

UAI 2008


Discussion
Discussion Derivative-Sum-Product Algorithm

  • 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

UAI 2008


Thanks
Thanks Derivative-Sum-Product Algorithm

  • Questions?

UAI 2008


Interpretation of skill updates
Interpretation of skill updates Derivative-Sum-Product Algorithm

  • For any given player let denote the outcomes of games he/she has played previously

  • Then the skill function corresponds to

UAI 2008


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