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416
An introduction to variational methods for graphical models
- TO APPEAR: M. I. JORDAN, (ED.), LEARNING IN GRAPHICAL MODELS
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Dynamic Bayesian Networks: Representation, Inference and Learning
, 2002
"... Modelling sequential data is important in many areas of science and engineering. Hidden Markov models (HMMs) and Kalman filter models (KFMs) are popular for this because they are simple and flexible. For example, HMMs have been used for speech recognition and bio-sequence analysis, and KFMs have bee ..."
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Cited by 758 (3 self)
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Modelling sequential data is important in many areas of science and engineering. Hidden Markov models (HMMs) and Kalman filter models (KFMs) are popular for this because they are simple and flexible. For example, HMMs have been used for speech recognition and bio-sequence analysis, and KFMs have been used for problems ranging from tracking planes and missiles to predicting the economy. However, HMMs
and KFMs are limited in their “expressive power”. Dynamic Bayesian Networks (DBNs) generalize HMMs by allowing the state space to be represented in factored form, instead of as a single discrete random variable. DBNs generalize KFMs by allowing arbitrary probability distributions, not just (unimodal) linear-Gaussian. In this thesis, I will discuss how to represent many different kinds of models as DBNs, how to perform exact and approximate inference in DBNs, and how to learn DBN models from sequential data.
In particular, the main novel technical contributions of this thesis are as follows: a way of representing
Hierarchical HMMs as DBNs, which enables inference to be done in O(T) time instead of O(T 3), where T is the length of the sequence; an exact smoothing algorithm that takes O(log T) space instead of O(T); a simple way of using the junction tree algorithm for online inference in DBNs; new complexity bounds on exact online inference in DBNs; a new deterministic approximate inference algorithm called factored frontier; an analysis of the relationship between the BK algorithm and loopy belief propagation; a way of
applying Rao-Blackwellised particle filtering to DBNs in general, and the SLAM (simultaneous localization
and mapping) problem in particular; a way of extending the structural EM algorithm to DBNs; and a variety of different applications of DBNs. However, perhaps the main value of the thesis is its catholic presentation of the field of sequential data modelling.
Good Error-Correcting Codes based on Very Sparse Matrices
, 1999
"... We study two families of error-correcting codes defined in terms of very sparse matrices. "MN" (MacKay--Neal) codes are recently invented, and "Gallager codes" were first investigated in 1962, but appear to have been largely forgotten, in spite of their excellent properties. The ..."
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Cited by 741 (23 self)
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We study two families of error-correcting codes defined in terms of very sparse matrices. "MN" (MacKay--Neal) codes are recently invented, and "Gallager codes" were first investigated in 1962, but appear to have been largely forgotten, in spite of their excellent properties. The decoding of both codes can be tackled with a practical sum-product algorithm. We prove that these codes are "very good," in that sequences of codes exist which, when optimally decoded, achieve information rates up to the Shannon limit. This result holds not only for the binary-symmetric channel but also for any channel with symmetric stationary ergodic noise. We give experimental results for binary-symmetric channels and Gaussian channels demonstrating that practical performance substantially better than that of standard convolutional and concatenated codes can be achieved; indeed, the performance of Gallager codes is almost as close to the Shannon limit as that of turbo codes.
Loopy Belief Propagation for Approximate Inference: An Empirical Study
- In Proceedings of Uncertainty in AI
, 1999
"... Recently, researchers have demonstrated that "loopy belief propagation" --- the use of Pearl's polytree algorithm in a Bayesian network with loops --- can perform well in the context of error-correcting codes. The most dramatic instance of this is the near Shannon-limit performa ..."
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Cited by 680 (18 self)
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Recently, researchers have demonstrated that "loopy belief propagation" --- the use of Pearl's polytree algorithm in a Bayesian network with loops --- can perform well in the context of error-correcting codes. The most dramatic instance of this is the near Shannon-limit performance of "Turbo Codes" --- codes whose decoding algorithm is equivalent to loopy belief propagation in a chain-structured Bayesian network. In this paper we ask: is there something special about the error-correcting code context, or does loopy propagation work as an approximate inference scheme in a more general setting? We compare the marginals computed using loopy propagation to the exact ones in four Bayesian network architectures, including two real-world networks: ALARM and QMR. We find that the loopy beliefs often converge and when they do, they give a good approximation to the correct marginals. However, on the QMR network, the loopy beliefs oscillated and had no obvious relationship ...
Learning probabilistic relational models
- In IJCAI
, 1999
"... A large portion of real-world data is stored in commercial relational database systems. In contrast, most statistical learning methods work only with "flat " data representations. Thus, to apply these methods, we are forced to convert our data into a flat form, thereby losing much ..."
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Cited by 619 (31 self)
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A large portion of real-world data is stored in commercial relational database systems. In contrast, most statistical learning methods work only with "flat " data representations. Thus, to apply these methods, we are forced to convert our data into a flat form, thereby losing much of the relational structure present in our database. This paper builds on the recent work on probabilistic relational models (PRMs), and describes how to learn them from databases. PRMs allow the properties of an object to depend probabilistically both on other properties of that object and on properties of related objects. Although PRMs are significantly more expressive than standard models, such as Bayesian networks, we show how to extend well-known statistical methods for learning Bayesian networks to learn these models. We describe both parameter estimation and structure learning — the automatic induction of the dependency structure in a model. Moreover, we show how the learning procedure can exploit standard database retrieval techniques for efficient learning from large datasets. We present experimental results on both real and synthetic relational databases. 1
Constructing Free Energy Approximations and Generalized Belief Propagation Algorithms
- IEEE Transactions on Information Theory
, 2005
"... Important inference problems in statistical physics, computer vision, error-correcting coding theory, and artificial intelligence can all be reformulated as the computation of marginal probabilities on factor graphs. The belief propagation (BP) algorithm is an efficient way to solve these problems t ..."
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Cited by 586 (13 self)
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Important inference problems in statistical physics, computer vision, error-correcting coding theory, and artificial intelligence can all be reformulated as the computation of marginal probabilities on factor graphs. The belief propagation (BP) algorithm is an efficient way to solve these problems that is exact when the factor graph is a tree, but only approximate when the factor graph has cycles. We show that BP fixed points correspond to the stationary points of the Bethe approximation of the free energy for a factor graph. We explain how to obtain regionbased free energy approximations that improve the Bethe approximation, and corresponding generalized belief propagation (GBP) algorithms. We emphasize the conditions a free energy approximation must satisfy in order to be a “valid ” or “maxent-normal ” approximation. We describe the relationship between four different methods that can be used to generate valid approximations: the “Bethe method, ” the “junction graph method, ” the “cluster variation method, ” and the “region graph method.” Finally, we explain how to tell whether a region-based approximation, and its corresponding GBP algorithm, is likely to be accurate, and describe empirical results showing that GBP can significantly outperform BP.
Learning low-level vision
- International Journal of Computer Vision
, 2000
"... We show a learning-based method for low-level vision problems. We set-up a Markov network of patches of the image and the underlying scene. A factorization approximation allows us to easily learn the parameters of the Markov network from synthetic examples of image/scene pairs, and to e ciently prop ..."
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Cited by 586 (31 self)
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We show a learning-based method for low-level vision problems. We set-up a Markov network of patches of the image and the underlying scene. A factorization approximation allows us to easily learn the parameters of the Markov network from synthetic examples of image/scene pairs, and to e ciently propagate image information. Monte Carlo simulations justify this approximation. We apply this to the \super-resolution " problem (estimating high frequency details from a low-resolution image), showing good results. For the motion estimation problem, we show resolution of the aperture problem and lling-in arising from application of the same probabilistic machinery.
Belief Propagation
, 2010
"... When a pair of nuclear-powered Russian submarines was reported patrolling off the eastern seaboard of the U.S. last summer, Pentagon officials expressed wariness over the Kremlin’s motivations. At the same time, these officials emphasized their confidence in the U.S. Navy’s tracking capabilities: “W ..."
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Cited by 479 (11 self)
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When a pair of nuclear-powered Russian submarines was reported patrolling off the eastern seaboard of the U.S. last summer, Pentagon officials expressed wariness over the Kremlin’s motivations. At the same time, these officials emphasized their confidence in the U.S. Navy’s tracking capabilities: “We’ve known where they were,” a senior Defense Department official told the New York Times, “and we’re not concerned about our ability to track the subs.” While the official did not divulge the methods used by the Navy to track submarines, the Times added that such
The generalized distributive law
- Information Theory, IEEE Transactions on
"... Abstract—In this semitutorial paper we discuss a general message passing algorithm, which we call the generalized dis-tributive law (GDL). The GDL is a synthesis of the work of many authors in the information theory, digital communications, signal processing, statistics, and artificial intelligence ..."
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Cited by 364 (2 self)
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Abstract—In this semitutorial paper we discuss a general message passing algorithm, which we call the generalized dis-tributive law (GDL). The GDL is a synthesis of the work of many authors in the information theory, digital communications, signal processing, statistics, and artificial intelligence communities. It includes as special cases the Baum–Welch algorithm, the fast Fourier transform (FFT) on any finite Abelian group, the Gal-lager–Tanner–Wiberg decoding algorithm, Viterbi’s algorithm, the BCJR algorithm, Pearl’s “belief propagation ” algorithm, the Shafer–Shenoy probability propagation algorithm, and the turbo decoding algorithm. Although this algorithm is guaranteed to give exact answers only in certain cases (the “junction tree ” condition), unfortunately not including the cases of GTW with cycles or turbo decoding, there is much experimental evidence, and a few theorems, suggesting that it often works approximately even when it is not supposed to. Index Terms—Belief propagation, distributive law, graphical models, junction trees, turbo codes. I.
The Bayes Net Toolbox for MATLAB
- Computing Science and Statistics
, 2001
"... The Bayes Net Toolbox (BNT) is an open-source Matlab package for directed graphical models. BNT supports many kinds of nodes (probability distributions), exact and approximate inference, parameter and structure learning, and static and dynamic models. BNT is widely used in teaching and research: the ..."
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Cited by 243 (1 self)
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The Bayes Net Toolbox (BNT) is an open-source Matlab package for directed graphical models. BNT supports many kinds of nodes (probability distributions), exact and approximate inference, parameter and structure learning, and static and dynamic models. BNT is widely used in teaching and research: the web page has received over 28,000 hits since May 2000. In this paper, we discuss a broad spectrum of issues related to graphical models (directed and undirected), and describe, at a high-level, how BNT was designed to cope with them all. We also compare BNT to other software packages for graphical models, and to the nascent OpenBayes effort.
