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10
Graphical models, exponential families, and variational inference
, 2008
"... The formalism of probabilistic graphical models provides a unifying framework for capturing complex dependencies among random variables, and building largescale multivariate statistical models. Graphical models have become a focus of research in many statistical, computational and mathematical fiel ..."
Abstract

Cited by 453 (25 self)
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The formalism of probabilistic graphical models provides a unifying framework for capturing complex dependencies among random variables, and building largescale multivariate statistical models. Graphical models have become a focus of research in many statistical, computational and mathematical fields, including bioinformatics, communication theory, statistical physics, combinatorial optimization, signal and image processing, information retrieval and statistical machine learning. Many problems that arise in specific instances — including the key problems of computing marginals and modes of probability distributions — are best studied in the general setting. Working with exponential family representations, and exploiting the conjugate duality between the cumulant function and the entropy for exponential families, we develop general variational representations of the problems of computing likelihoods, marginal probabilities and most probable configurations. We describe how a wide varietyof algorithms — among them sumproduct, cluster variational methods, expectationpropagation, mean field methods, maxproduct and linear programming relaxation, as well as conic programming relaxations — can all be understood in terms of exact or approximate forms of these variational representations. The variational approach provides a complementary alternative to Markov chain Monte Carlo as a general source of approximation methods for inference in largescale statistical models.
TreeBased Reparameterization Framework for Analysis of Belief Propagation and Related Algorithms
, 2001
"... We present a treebased reparameterization framework that provides a new conceptual view of a large class of algorithms for computing approximate marginals in graphs with cycles. This class includes the belief propagation or sumproduct algorithm [39, 36], as well as a rich set of variations and ext ..."
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Cited by 102 (21 self)
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We present a treebased reparameterization framework that provides a new conceptual view of a large class of algorithms for computing approximate marginals in graphs with cycles. This class includes the belief propagation or sumproduct algorithm [39, 36], as well as a rich set of variations and extensions of belief propagation. Algorithms in this class can be formulated as a sequence of reparameterization updates, each of which entails refactorizing a portion of the distribution corresponding to an acyclic subgraph (i.e., a tree). The ultimate goal is to obtain an alternative but equivalent factorization using functions that represent (exact or approximate) marginal distributions on cliques of the graph. Our framework highlights an important property of BP and the entire class of reparameterization algorithms: the distribution on the full graph is not changed. The perspective of treebased updates gives rise to a simple and intuitive characterization of the fixed points in terms of tree consistency. We develop interpretations of these results in terms of information geometry. The invariance of the distribution, in conjunction with the fixed point characterization, enables us to derive an exact relation between the exact marginals on an arbitrary graph with cycles, and the approximations provided by belief propagation, and more broadly, any algorithm that minimizes the Bethe free energy. We also develop bounds on this approximation error, which illuminate the conditions that govern their accuracy. Finally, we show how the reparameterization perspective extends naturally to more structured approximations (e.g., Kikuchi and variants [52, 37]) that operate over higher order cliques.
A New Look at Survey Propagation and its Generalizations
"... We study the survey propagation algorithm [19, 5, 4], which is an iterative technique that appears to be very effective in solving random kSAT problems even with densities close to threshold. We first describe how any SAT formula can be associated with a novel family of Markov random fields (MRFs), ..."
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Cited by 49 (13 self)
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We study the survey propagation algorithm [19, 5, 4], which is an iterative technique that appears to be very effective in solving random kSAT problems even with densities close to threshold. We first describe how any SAT formula can be associated with a novel family of Markov random fields (MRFs), parameterized by a real number ρ. We then show that applying belief propagation— a wellknown “messagepassing” technique—to this family of MRFs recovers various algorithms, ranging from pure survey propagation at one extreme (ρ = 1) to standard belief propagation on the uniform distribution over SAT assignments at the other extreme (ρ = 0). Configurations in these MRFs have a natural interpretation as generalized satisfiability assignments, on which a partial order can be defined. We isolate cores as minimal elements in this partial
TreeBased Reparameterization for Approximate Estimation on Loopy Graphs
 Advances in Neural Information Processing Systems (NIPS
, 2001
"... We present a treebased reparameterization framework that provides a new conceptual view of a large class of iterative algorithms for computing approximate marginals in graphs with cycles. It includes belief propagation (BP), which can be reformulated as a very local form of reparameterization. Mor ..."
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Cited by 49 (4 self)
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We present a treebased reparameterization framework that provides a new conceptual view of a large class of iterative algorithms for computing approximate marginals in graphs with cycles. It includes belief propagation (BP), which can be reformulated as a very local form of reparameterization. More generally, we consider algorithms that perform exact computations over spanning trees of the full graph. On the practical side, we nd that such tree reparameterization (TRP) algorithms typically converge more quickly than BP with lower cost per iteration; moreover, TRP often converges on problems for which BP fails. The reparameterization perspective also provides theoretical insight into approximate estimation, including a new probabilistic characterization of xed points; and an invariance intrinsic to TRP/BP. These two properties in conjunction enable us to analyze and bound the approximation error that arises in applying these techniques. Our results also have natural extensions to approximations (e.g., Kikuchi) that involve clustering nodes. 1
The DLR Hierarchy of Approximate Inference
"... We propose a hierarchy for approximate inference based on the Dobrushin, Lanford, Ruelle (DLR) equations. This hierarchy includes existing algorithms, such as belief propagation, and also motivates novel algorithms such as factorized neighbors (FN) algorithms and variants of mean field (MF) al ..."
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Cited by 8 (1 self)
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We propose a hierarchy for approximate inference based on the Dobrushin, Lanford, Ruelle (DLR) equations. This hierarchy includes existing algorithms, such as belief propagation, and also motivates novel algorithms such as factorized neighbors (FN) algorithms and variants of mean field (MF) algorithms.
Convergence analysis of reweighted sumproduct algorithms
 In Int. Conf. Acoustic, Speech and Sig. Proc
, 2007
"... Abstract—Markov random fields are designed to represent structured dependencies among large collections of random variables, and are wellsuited to capture the structure of realworld signals. Many fundamental tasks in signal processing (e.g., smoothing, denoising, segmentation etc.) require efficie ..."
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Cited by 8 (3 self)
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Abstract—Markov random fields are designed to represent structured dependencies among large collections of random variables, and are wellsuited to capture the structure of realworld signals. Many fundamental tasks in signal processing (e.g., smoothing, denoising, segmentation etc.) require efficient methods for computing (approximate) marginal probabilities over subsets of nodes in the graph. The marginalization problem, though solvable in linear time for graphs without cycles, is computationally intractable for general graphs with cycles. This intractability motivates the use of approximate “messagepassing ” algorithms. This paper studies the convergence and stability properties of the family of reweighted sumproduct algorithms, a generalization of the widely used sumproduct or belief propagation algorithm, in which messages are adjusted with graphdependent weights. For pairwise Markov random fields, we derive various conditions that are sufficient to ensure convergence, and also provide bounds on the geometric convergence rates. When specialized to the ordinary sumproduct algorithm, these results provide strengthening of previous analyses. We prove that some of our conditions are necessary and sufficient for subclasses of homogeneous models, but not for general models. The experimental simulations on various classes of graphs validate our theoretical results. Index Terms—Approximate marginalization, belief propagation, convergence analysis, graphical models, Markov random fields, sumproduct algorithm. I.
TreeBased Reparameterization Framework for Approximate Estimation of Stochastic Processes on Graphs With Cycles
, 2001
"... We present a treebased reparameterization framework for the approximate estimation of stochastic processes on graphs with cycles. This framework provides a new conceptual view of a large class of iterative algorithms for computing approximate marginals in graphs with cycles. Among them is belief pr ..."
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Cited by 6 (2 self)
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We present a treebased reparameterization framework for the approximate estimation of stochastic processes on graphs with cycles. This framework provides a new conceptual view of a large class of iterative algorithms for computing approximate marginals in graphs with cycles. Among them is belief propagation (BP), otherwise known as the sumproduct algorithm, which can be reformulated as a very local form of reparameterization. More generally, this class includes algorithms in which updates are more global, and involve performing exact computations over spanning trees of the full graph. On the practical side, we nd that such tree reparameterization (TRP) algorithms typically converge more quickly than belief propagation with equivalent or lower cost per iteration; moreover, TRP often converges on harder problems for which BP fails. The reparameterization perspective also provides new theoretical insights into approximate estimation. In particular, it leads to a novel probabilistic characterization of the set of xed points. We develop the geometry of treebased reparameterization, and use it to develop sucient conditions for convergence in the case of two spanning trees. A fundamental property of reparameterization updates is that they leave invariant the distribution on the full graph. This invariance, in conjunction with our xed point characterization, enables us to derive an exact relation between the true marginals on an arbitrary graph with cycles, and the approximations provided by TRP or BP. We also develop bounds on this approximation error, which illuminate the conditions that govern performance of such techniques. Our results also have natural extensions to approximations (e.g., Kikuchi) that involve clustering nodes. MW was supported in part by a 1967 Fel...
1 The DLR Hierarchy of Approximate Inference
"... We propose a hierarchy for approximate inference based on the Dobrushin, Lanford, Ruelle (DLR) equations. This hierarchy includes existing algorithms, such as belief propagation, and also motivates novel algorithms such as factorized neighbors (FN) algorithms and variants of mean field (MF) algorith ..."
Abstract
 Add to MetaCart
We propose a hierarchy for approximate inference based on the Dobrushin, Lanford, Ruelle (DLR) equations. This hierarchy includes existing algorithms, such as belief propagation, and also motivates novel algorithms such as factorized neighbors (FN) algorithms and variants of mean field (MF) algorithms. In particular, we show that extrema of the Bethe free energy correspond to approximate solutions of the DLR equations. In addition, we demonstrate a close connection between these approximate algorithms and Gibbs sampling. Finally, we compare and contrast various of the algorithms in the DLR hierarchy on spinglass problems. The experiments show that algorithms higher up in the hierarchy give more accurate results when they converge but tend to be less stable. 1