12/07/2008UAI 2008 Cumulative Distribution Networks and the Derivative-Sum-Product Algorithm Jim C. Huang and Brendan J. Frey Probabilistic and Statistical.

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Presentation transcript:

12/07/2008UAI 2008 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

12/07/2008UAI 2008 Problems where density models may be intractable e.g.: Modelling arbitrary dependencies e.g.: Modelling stochastic orderings Cumulative distribution network (CDN) Motivation e.g.: Predicting game outcomes in Halo 2

12/07/2008UAI 2008 Cumulative distribution networks (CDNs) Graphical model of the cumulative distribution function (CDF) Example:

12/07/2008UAI 2008 Positive convergence Negative convergence Monotonicity Cumulative distribution functions Marginalization  maximization Conditioning  differentiation

12/07/2008UAI 2008 Necessary/sufficient conditions on CDN functions Negative convergence (necessity and sufficiency): Positive convergence (sufficiency): For each X k, at least one neighboring function  0 All functions  1

12/07/2008UAI 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

12/07/2008UAI 2008 Marginal independence Marginalization  maximization –e.g.: X is marginally independent of Y

12/07/2008UAI 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

12/07/2008UAI 2008 Check: A toy example Markov random fields Required “Bayes net”

12/07/2008UAI 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) …

12/07/2008UAI 2008 Derivative-sum-product In a CDN: In a factor graph:

12/07/2008UAI 2008 Ranking in multiplayer gaming e.g.: Halo 2 game with 7 players, 3 teams Player skill functions Player performanc e Team performanc e Given game outcomes, update player skills as a function of all player/team performances

12/07/2008UAI 2008 Ranking in multiplayer gaming = Local cumulative model linking team rank r n with player performances x n e.g.: Team 2 has rank 2

12/07/2008UAI 2008 Ranking in multiplayer gaming Enforce stochastic orderings between teams via h = Pairwise model of team ranks r n,r n+1

12/07/2008UAI 2008 CDN functions = Gaussian CDFs Skill updates: Prediction: Ranking in multiplayer gaming

12/07/2008UAI 2008 Results Previous methods for ranking players: –ELO (Elo, 1978) –TrueSkill (Graepel, Minka and Herbrich, 2006) After message-passing…

12/07/2008UAI 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

12/07/2008UAI 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 –…

12/07/2008UAI 2008 Thanks Questions?

12/07/2008UAI 2008 Null-dependence in CDNs Given that X s, X t, X u are sets of discrete variables over the alphabet A =  0,…,K , if X s, X t are marginally independent and X u = (0,…,0), then X s and X t are conditionally independent given X u Proof:

12/07/2008UAI 2008 The CDN as a random field Take an undirected graph G = (V,E) (NOT an MRF/CRF!) Define CUT(A,B) as a set of nodes which if removed, cuts G into two parts containing A,B If the random field satisfies: Necessary and sufficient for potentials to be defined over connected components of G (proof similar to that of Hammersley-Clifford theorem)

12/07/2008UAI 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

12/07/2008UAI 2008 Derivative-Sum-Product Message from function to variable: …

12/07/2008UAI 2008 Derivative-Sum-Product Message from variable to function: …