cumulative distribution networks and the derivative sum product algorithm
Download
Skip this Video
Download Presentation
Cumulative Distribution Networks and the Derivative-Sum-Product Algorithm

Loading in 2 Seconds...

play fullscreen
1 / 20

Cumulative Distribution Networks and the Derivative-Sum-Product Algorithm - PowerPoint PPT Presentation


  • 66 Views
  • Uploaded on

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

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Cumulative Distribution Networks and the Derivative-Sum-Product Algorithm' - iola


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
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

ad