Cumulative distribution networks and the derivative sum product algorithm
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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

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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

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)

  • Graphical model of the cumulative distribution function (CDF)

  • Example:

UAI 2008


Cumulative distribution functions

Negative convergence

  • Marginalization  maximization

  • Conditioning  differentiation

Positive convergence

Monotonicity

UAI 2008


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

UAI 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

UAI 2008


Marginal independence

  • Marginalization  maximization

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

UAI 2008


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

UAI 2008


A toy example

Required “Bayes net”

Markov random fields

Check:

UAI 2008


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)

UAI 2008


Derivative-sum-product

  • In a CDN:

  • In a factor graph:

UAI 2008


Ranking in multiplayer gaming

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 gaming

= Local cumulative model linking team rank rn

with player performances xn

e.g.: Team 2 has rank 2

UAI 2008


Ranking in multiplayer gaming

= Pairwise model of team ranks rn,rn+1

Enforce stochastic orderings between teams via h

UAI 2008


Ranking in multiplayer gaming

  • CDN functions = Gaussian CDFs

  • Skill updates:

  • Prediction:

UAI 2008


Results

  • Previous methods for ranking players:

    • ELO (Elo, 1978)

    • TrueSkill (Graepel, Minka and Herbrich, 2006)

  • After message-passing…

UAI 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

UAI 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

UAI 2008


Thanks

  • Questions?

UAI 2008


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

UAI 2008


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