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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

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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