Major Advances and Emerging Developments of Graphical Models
Michael I. Jordan, Erik B. Sudderth, Martin J. Wainwright, Alan S. Willsky · 2010
Graphical models, referred to in various guises as Markov random fields (MRFs), Bayesian networks, factor graphs, influence diagrams, decision networks, or structured stochastic systems, are a powerful and elegant marriage of graph theory, probability theory, and decision theory. They yield a unifying perspective on many long-standing and emerging frameworks for modeling complex phenomena, as well as methods for processing complex sources of data and signals. Such models are of particular importance in areas of signal processing that overlap with machine learning, time-series analysis, spatial statistics, and optimization. Graphical representations of signals have a long history in signal processing. For example, signal flow diagrams, allpole/autoregressive modeling, MRF models for images, and pyramidal/ multiresolution signal representations are well-known instances of graphbased models. In each of these and many other cases, it is the exploitation of the graphical structure of these models and more specifically the interplay of that structure and the probabilistic and/or dynamic behavior of the phenomenon being modeled that leads to both powerful algorithms and conceptual insights. While the use of graphical representations has a long and successful history in signal processing, major advances in modeling and processing methodologies are continually emerging. This is due in large part to the growing interdisciplinary nature of this line of inquiry, which attracts researchers in fields ranging from nonparametric statistics to statistical physics, as well as machine learning, optimization, and signal processing. The purpose of this special issue is to introduce some of these emerging developments, highlight their implications and applications in signal processing, and provide insights into a rich field where statistics and engineering come together. One of the key connections between graphical models and signal processing is the class of hidden Markov models (HMMs) and other state-space models. Indeed, the graphs underlying these models take the form of chains, and the graphical model formalism can be viewed as an extension of the Markovian concepts underlying state-space models to general graphs. The special issue accordingly begins with a tutorial article by Barber and Cemgil that discusses graphical representations for HMMs and linear dynamical systems. The article also discusses switching state-space models, which are time-series models that have both discrete and continuous components. These models have a wide range of signal processing applications. The subsequent article by Bilmes provides further illustration of the way in which graphical modeling ideas can be used to