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The junction tree algorithm provides a systematic solution to the problem of computing likelihoods and other statistical quantities associated with a graphical model

Graphical Models, Exponential Families, and Variational Inference

Graphical Models, Exponential Families, and Variational Inference, no. 1–2 (2008): 1-305

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摘要

The formalism of probabilistic graphical models provides a unifying framework for capturing complex dependencies among random variables, and building large-scale multivariate statistical models. Graphical models have become a focus of research in many statistical, computational and mathematical fields, including bioinformatics, communicat...更多

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简介
  • Introduction complexity and feasibility

    In particular, the running time of an algorithm or the magnitude of an error bound can often be characterized in terms of structural properties of a graph.
  • “A linear programming formulation and approximation algorithms for the metric labeling problem,” SIAM Journal on Discrete Mathematics, vol 18, no.
  • J. Wainwright, “Probabilistic analysis of linear programming decoding,” IEEE Transactions on Information Theory, vol 54, no.
  • “Convergent message-passing algorithms for inference over general graphs with convex free energy,” in Proceedings of the 24th Conference on Uncertainty in Artificial Intelligence, Arlington, VA: AUAI Press, 2008.
重点内容
  • Introduction complexity and feasibility

    In particular, the running time of an algorithm or the magnitude of an error bound can often be characterized in terms of structural properties of a graph
  • As we discuss, the computational complexity of a fundamental method known as the junction tree algorithm — which generalizes many of the recursive algorithms on graphs cited above — can be characterized in terms of a natural graphtheoretic measure of interaction among variables
  • The junction tree algorithm provides a systematic solution to the problem of computing likelihoods and other statistical quantities associated with a graphical model
  • One popular source of methods for attempting to cope with such cases is the Markov chain Monte Carlo (MCMC) framework, and there is a significant literature on the application of Markov chain Monte Carlo methods to graphical models [e.g., 28, 93, 202]. Our focus in this survey is rather different: we present an alternative computational methodology for statistical inference that is based on variational methods
  • These techniques provide a general class of alternatives to Markov chain Monte Carlo, and have applications outside of the graphical model framework
  • The principal object of interest in our exposition is a certain conjugate dual relation associated with exponential families. From this foundation of conjugate duality, we develop a general variational representation for computing likelihoods and marginal probabilities in exponential families
结果
  • “Factor graphs and the sum-product algorithm,” IEEE Transactions on Information Theory, vol 47, no.
  • J. Spiegelhalter, “Local computations with probabilities on graphical structures and their application to expert systems,” Journal of the Royal Statistical Society, Series B, vol 50, pp.
  • S. Willsky, “Walk-sums and belief propagation in Gaussian graphical models,” Journal of Machine Learning Research, vol 7, pp.
  • “Bayesian inference and optimal design in the sparse linear model,” in Proceedings of the Eleventh International Conference on Artificial Intelligence and Statistics, San Juan, Puerto Rico, 2007.
  • S. Willsky, “Loop series and Bethe variational bounds for attractive graphical models,” in Advances in Neural Information Processing Systems, pp.
  • “Comparative study of energy minimization methods for Markov random fields,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol 30, pp.
  • S. Willsky, “Tree-based reparameterization framework for analysis of sum-product and related algorithms,” IEEE Transactions on Information Theory, vol 49, no.
  • S. Willsky, “Tree-reweighted belief propagation algorithms and approximate ML, estimation by pseudomoment matching,” in Proceedings of the Ninth International Conference on Artificial Intelligence and Statistics, 2003.
  • S. Willsky, “Exact MAP estimates via agreement ontrees: Linear programming and message-passing,” IEEE Transactions on Information Theory, vol 51, no.
  • S. Willsky, “A new class of upper bounds on the log partition function,” IEEE Transactions on Information Theory, vol 51, no.
  • I. Jordan, “Log-determinant relaxation for approximate inference in discrete Markov random fields,” IEEE Transactions on Signal Processing, vol 54, no.
结论
  • T. Freeman, “Correctness of belief propagation in Gaussian graphical models of arbitrary topology,” in Advances in Neural Information Processing Systems, pp.
  • “MAP estimation, linear programming, and belief propagation with convex free energies,” in Proceedings of the 23rd Conference on Uncertainty in Artificial Intelligence, Arlington, VA: AUAI Press, 2007.
  • “A linear programming approach to max-sum problem: A review,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol 29, no.
  • Wiegerinck, “Variational approximations between mean field theory and the junction tree algorithm,” in Proceedings of the 16th Conference on Uncertainty in Artificial Intelligence, pp.
基金
  • In the USA: This journal is registered at the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923
  • Authorization does not extend to other kinds of copying, such as that for general distribution, for advertising or promotional purposes, for creating new collective works, or for resale
  • In the rest of the world: Permission to photocopy must be obtained from the copyright owner
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