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

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

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)

  • Graphical model of the cumulative distribution function (CDF)

  • Example:

UAI 2008


Cumulative distribution functions

Cumulative distribution functions

Negative convergence

  • Marginalization  maximization

  • Conditioning  differentiation

Positive convergence

Monotonicity

UAI 2008


Necessary sufficient conditions on cdn functions

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 functions1

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

Marginal independence

  • Marginalization  maximization

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

UAI 2008


Conditional independence

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

A toy example

Required “Bayes net”

Markov random fields

Check:

UAI 2008


Inference by message passing

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

Derivative-sum-product

  • In a CDN:

  • In a factor graph:

UAI 2008


Ranking in multiplayer gaming

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 gaming1

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 gaming2

Ranking in multiplayer gaming

= 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

  • CDN functions = Gaussian CDFs

  • Skill updates:

  • Prediction:

UAI 2008


Results

Results

  • Previous methods for ranking players:

    • ELO (Elo, 1978)

    • TrueSkill (Graepel, Minka and Herbrich, 2006)

  • After message-passing…

UAI 2008


Summary

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

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

Thanks

  • Questions?

UAI 2008


Interpretation of skill updates

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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