Entropy of Hidden Markov Processes and Connections to Dynamical Systems Book contents
- Frontmatter
- Contents
- Introduction
- 1 Hidden Markov processes in the context of symbolic dynamics
- 2 On the preservation of Gibbsianness under symbol amalgamation
- 3 A note on a complex Hilbert metric with application to domain of analyticity for entropy rate of hidden Markov processes
- 4 Bounds on the entropy rate of binary hidden Markov processes
- 5 Entropy rate for hidden Markov chains with rare transitions
- 6 The capacity of finite-state channels in the high-noise regime
- 7 Computing entropy rates for hidden Markov processes
- 8 Factors of Gibbs measures for full shifts
- 9 Thermodynamics of hidden Markov processes
1 - Hidden Markov processes in the context of symbolic dynamics
Published online by Cambridge University Press: 05 June 2011
- Frontmatter
- Contents
- Introduction
- 1 Hidden Markov processes in the context of symbolic dynamics
- 2 On the preservation of Gibbsianness under symbol amalgamation
- 3 A note on a complex Hilbert metric with application to domain of analyticity for entropy rate of hidden Markov processes
- 4 Bounds on the entropy rate of binary hidden Markov processes
- 5 Entropy rate for hidden Markov chains with rare transitions
- 6 The capacity of finite-state channels in the high-noise regime
- 7 Computing entropy rates for hidden Markov processes
- 8 Factors of Gibbs measures for full shifts
- 9 Thermodynamics of hidden Markov processes
Summary
Abstract. In an effort to aid communication among different fields and perhaps facilitate progress on problems common to all of them, this article discusses hidden Markov processes from several viewpoints, especially that of symbolic dynamics, where they are known as sofic measures or continuous shift-commuting images of Markov measures. It provides background, describes known tools and methods, surveys some of the literature, and proposes several open problems.
Introduction
Symbolic dynamics is the study of shift (and other) transformations on spaces of infinite sequences or arrays of symbols and maps between such systems. A symbolic dynamical system, with a shift-invariant measure, corresponds to a stationary stochastic process. In the setting of information theory, such a system amounts to a collection of messages. Markov measures and hidden Markov measures, also called sofic measures, on symbolic dynamical systems have the desirable property of being determined by a finite set of data. But not all of their properties, for example the entropy, can be determined by finite algorithms. This article surveys some of the known and unknown properties of hidden Markov measures that are of special interest from the viewpoint of symbolic dynamics. To keep the article self contained, necessary background and related concepts are reviewed briefly. More can be found in [47, 56, 55, 71].
We discuss methods and tools that have been useful in the study of symbolic systems, measures supported on them, and maps between them.
- Type
- Chapter
- Information
- Entropy of Hidden Markov Processes and Connections to Dynamical SystemsPapers from the Banff International Research Station Workshop, pp. 5 - 71Publisher: Cambridge University PressPrint publication year: 2011
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