We introduce a new class of problems called network information flow which is inspired by computer network applications. Consider a point-to-point communication network on which a number of information sources are to be mulitcast to certain sets of destinations. We assume that the information sources are mutually independent. The problem is to characterize the admissible coding rate region. This model subsumes all previously studied models along the same line. In this paper, we study the problem with one information source, and we have obtained a simple characterization of the admissible coding rate region. Our result can be regarded as the Max-flow Min-cut Theorem for network information flow. Contrary to one’s intuition, our work reveals that it is in general not optimal to regard the information to be multicast as a “fluid” which can simply be routed or replicated. Rather, by employing coding at the nodes, which we refer to as network coding, bandwidth can in general be saved. This finding may have significant impact on future design of switching systems.

Statistical applications in fields such as bioinformatics, information retrieval, speech processing, image processing and communications often involve large-scale models in which thousands or millions of random variables are linked in complex ways. Graphical models provide a general methodology for approaching these problems, and indeed many of the models developed by researchers in these applied fields are instances of the general graphical model formalism. We review some of the basic ideas underlying graphical models, including the algorithmic ideas that allow graphical models to be deployed in large-scale data analysis problems. We also present examples of graphical models in bioinformatics, error-control coding and language processing. Key words and phrases: Probabilistic graphical models, junction tree algorithm, sum-product algorithm, Markov chain Monte Carlo, variational inference, bioinformatics, error-control coding.

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.

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.

In applications of graphical models arising in fields such as computer vision, the hidden variables of interest are most naturally specified by continuous, non--Gaussian distributions. However, due to the limitations of existing inf#6F6F3 algorithms, it is of#]k necessary tof#3# coarse, discrete approximations to such models. In this paper, we develop a nonparametric belief propagation (NBP) algorithm, which uses stochastic methods to propagate kernel--based approximations to the true continuous messages. Each NBP message update is based on an efficient sampling procedure which can accomodate an extremely broad class of potentialf#l3]k[[z3 allowing easy adaptation to new application areas. We validate our method using comparisons to continuous BP for Gaussian networks, and an application to the stereo vision problem.

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.

Graphical models, suchasBayesian networks and Markov random fields, represent statistical dependencies of variables by a graph. The max-product "belief propagation" algorithm is a local-message passing algorithm on this graph that is known to converge to a unique fixed point when the graph is a tree. Furthermore, when the graph is a tree, the assignment based on the fixed-point yields the most probable a posteriori (MAP) values of the unobserved variables given the observed ones. Recently, good

In this work we show how to combine bottom-up and topdown approaches into a single figure-ground segmentation process. This process provides accurate delineation of object boundaries that cannot be achieved by either the topdown or bottom-up approach alone. The top-down approach uses object representation learned from examples to detect an object in a given input image and provide an approximation to its figure-ground segmentation. The bottomup approach uses image-based criteria to define coherent groups of pixels that are likely to belong together to either the figure or the background part. The combination provides a final segmentation that draws on the relative merits of both approaches: The result is as close as possible to the top-down approximation, but is also constrained by the bottom-up process to be consistent with significant image discontinuities. We construct a global cost function that represents these top-down and bottom-up requirements. We then show how the global minimum of this function can be efficiently found by applying the sum-product algorithm. This algorithm also provides a confidence map that can be used to identify image regions where additional top-down or bottom-up information may further improve the segmentation. Our experiments show that the results derived from the algorithm are superior to results given by a pure top-down or pure bottom-up approach. The scheme has broad applicability, enabling the combined use of a range of existing bottom-up and top-down segmentations. 1.

We present in this paper a new, complete method for distributed constraint optimization, based on dynamic programming. It is a utility propagation method, inspired by the sum-product algorithm, which is correct only for tree-shaped constraint networks. In this paper, we show how to extend that algorithm to arbitrary topologies using a pseudotree arrangement of the problem graph. Our algorithm requires a linear number of messages, whose maximal size depends on the induced width along the particular pseudotree chosen. We compare our algorithm with backtracking algorithms, and present experimental results. For some problem types we report orders of magnitude fewer messages, and the ability to deal with arbitrarily large problems. Our algorithm is formulated for optimization problems, but can be easily applied to satisfaction problems as well.

This paper presents a geometric approach to the construction of low-density parity-check (LDPC) codes. Four classes of LDPC codes are constructed based on the lines and points of Euclidean and projective geometries over finite fields. Codes of these four classes have good minimum distances and their Tanner graphs have girth T. Finite-geometry LDPC codes can be decoded in various ways, ranging from low to high decoding complexity and from reasonably good to very good performance. They perform very well with iterative decoding. Furthermore, they can be put in either cyclic or quasi-cyclic form. Consequently, their encoding can be achieved in linear time and implemented with simple feedback shift registers. This advantage is not shared by other LDPC codes in general and is important in practice. Finite-geometry LDPC codes can be extended and shortened in various ways to obtain other good LDPC codes. Several techniques of extension and shortening are presented. Long extended finite-geometry LDPC codes have been constructed and they achieve a performance only a few tenths of a decibel away from the Shannon theoretical limit with iterative decoding.