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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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dl.acm.org dblp.uni-trier.de academic.microsoft.com

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variational approachgeneral variational representationvariational inferenceexponential familiesvariational representation更多(21+)

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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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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.

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In the rest of the world: Permission to photocopy must be obtained from the copyright owner

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