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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

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

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

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

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

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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