Cumulative distribution networks: Graphical models for cumulative distribution functions

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Cumulative distribution networks Graphical
models for cumulative distribution functions

• Jim C. Huang and Brendan J. Frey
• Probabilistic and Statistical Inference Group,
  
  Department of Electrical and Computer
  Engineering,
  University of Toronto,
  
  Toronto, ON, Canada

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Motivation

• Problems where density models may be
  intractable/unsuitable
• e.g. Models with latent variables
  unidentifiability, intractability
• e.g. Learning to rank
• Cumulative distribution network (CDN)

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Cumulative distribution functions (CDFs)
Negative convergence
Positive convergence
Monotonicity

• Marginalization ? maximization
• Conditioning ? differentiation

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Cumulative distribution networks (CDNs)

• Bipartite graph for
  representing CDFs
• Example
• Sufficient for to be CDFs (Huang and
  Frey, 2008)
• e.g. Multivariate Gaussian CDFs, multivariate
  sigmoids,
• copulas

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Necessary/sufficient conditions on CDN functions

• Negative convergence (necessity and sufficiency)
  
• Positive convergence (sufficiency)

For each node a, at least one neighboring
function ? 0
All functions ? 1
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Necessary/sufficient conditions on CDN functions

• Monotonicity lemma (sufficiency)
• (assuming derivatives exist!)

All functions monotonically non-decreasing
Sufficient condition for a valid joint CDF
Each CDN function can be a CDF of its
arguments
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Example Bivariate CDN distributions

• Using Gaussian bivariate CDFs

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Example Bivariate CDN distributions

• Using Gumbel copulas with marginal t-CDFs and
  Gaussian CDFs

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Conditional independence and graph separation in
CDNs

• For any disjoint variable node sets
  separated by with respect to

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Conditional independence in CDNs

• For any disjoint variable node sets
  separated by with respect to
• e.g. X and Y are conditionally dependent given Z
  
• e.g. X and Y are conditionally independent given
  Z

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Conditional independence and graph separation in
CDNs
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Connection to bi-directed graphs

• Graphs for representing marginal independence
• e.g.
• Covariance graphs (Kauermann, 1996)
• Binary models for marginal independence (Drton
  and Richardson, 2008)
• Factorial mixture models (Silva and Ghahramani,
  2009)

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Null-dependence in CDNs
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Null-dependence in CDNs
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Mapping between CDNs and factor graphs

• Equivalence between bi-directed graph and
  directed graph
• Equivalence between CDN and factor graph

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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 algorithm (Huang and Frey,
  2008)

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The derivative-sum-product algorithm

• In a CDN
• In a factor graph

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Derivative-Sum-Product

• Message from function to variable

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Derivative-Sum-Product

• Message from variable to function

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Application Ranking in multiplayer gaming

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

Given game outcomes, update player skills as a
function of all player/team performances
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Ranking in multiplayer gaming
Local cumulative model linking team rank rn
with player performances xn
e.g. Team 2 has rank 2
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Ranking in multiplayer gaming
Pairwise model of team ranks rn,rn1
Enforce stochastic orderings between teams via h
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Application Ranking in multiplayer gaming

• CDN functions Gaussian CDFs
• Skill updates
• Prediction

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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

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Results

• Previous methods for ranking players
• ELO (Elo, 1978)
• TrueSkill (Graepel, Minka and Herbrich, 2006)
• After message-passing

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Factor graph and CDN for multiplayer games
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Factor graph and CDN for multiplayer games
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Factor graph and CDN for multiplayer games
Dual factor graph
TrueSkill factor graph
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Learning to rank from observations

• GOAL Learn a ranking function which
  minimizes probability of misranking on
• test queries

Training data
Learning
Predict on test data
?
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Structured ranking learning

• Define structured loss functional as likelihood
  of generating order graphs
• Use stochastic gradients to minimize structured
  loss functional given independent observations

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Converting from an order graph to a CDN
Edge in order graph
Preference variable node in CDN
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Probabilistic models for rank data as CDNs
Plackett-Luce model
Bradley-Terry model
e.g. RankNet (Burges et al, 2005), RankMotif
(Chen, Hughes and Morris, 2007)
e.g. ListNet (Cao et al, 2007), ListMLE (Xia et
al., 2008)
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Ranking documents for information retrieval

• Loss functional
• Multivariate sigmoids

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Ranking documents for information retrieval

• Ranking function
• Nadaraya-Watson estimator with Gaussian kernel

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Ranking documents for information retrieval

• Performance metrics
• Precision
• Average precision
• Normalized Discounted Cumulative Gains (NDCG) for
  a ranked list of documents with labels r(j)

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Application Information retrieval

• OHSUMED dataset (LETOR 2.0)

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Application Information retrieval

• OHSUMED dataset (LETOR 2.0)

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Application Information retrieval

• OHSUMED dataset (LETOR 2.0)

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Application Computational systems biology
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Ranking transcription factor binding sites

• Learn from protein binding microarray data
  (Berger et al. 2006)

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Ranking transcription factor binding sites

• Ranking function depends on position weight
  matrix M

Probability of occurrence
Position
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Ranking transcription factor binding sites
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Ranking transcription factor binding sites

• Learn to rank microRNA targets using diverse
  datasets

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Ranking microRNA targets

• Combine quantitative features and sequence data
• Quantitative features can be obtained from
  diverse experimental data and computational
  prediction methods

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Ranking microRNA targets

• Feature extraction

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Ranking microRNA targets

• PITA score (Kertesz et al., 2007) measures degree
  to which microRNA target site is accessible due
  to RNA secondary structure

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Ranking microRNA targets

• MicroRNA activity is associated with decreased
  target mRNA and protein abundance

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Discussion

• Maximum-likelihood learning in CDNs
• Message-passing in graphs with loops
• Approximations for DSP messages in continuous
  models
• Refinements to structured ranking learning
  framework
• Optimization algorithms
• Choice of CDN functions
• Choice of ranking function
• Further applications
• Vision
• Collaborative prediction
• Genomics, proteomics, immunology
• Problems with partial ordering of variables