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This volume is a collection of papers on hidden Markov processes (HMPs) involving connections with symbolic dynamics and statistical mechanics. The subject was the focus of a five-day workshop held at the Banff International Research Station (BIRS) in October 2007, which brought together thirty mathematicians, computer scientists, and electrical engineers from institutions throughout the world. Most of the papers in this volume are based either on work presented at the workshop or on problems posed at the workshop.
From one point of view, an HMP is a stochastic process obtained as the noisy observation process of a finite-state Markov chain; a simple example is a binary Markov chain observed in binary symmetric noise, i.e., each symbol (0 or 1) in a binary state sequence generated by a two-state Markov chain may be flipped with some small probability, independently from time instant to time instant. In another (essentially equivalent) viewpoint, an HMP is a process obtained from a finite-state Markov chain by partitioning its state set into groups and completely “hiding” the distinction among states within each group; more precisely, there is a deterministic function on the states of the Markov chain, and the HMP is the process obtained by observing the sequences of function values rather than sequences of states (and hence such a process is sometimes called a “function of a Markov chain”).
HMPs are encountered in an enormous variety of applications involving phenomena observed in the presence of noise.
Hidden Markov processes (HMPs) are important objects of study in many areas of pure and applied mathematics, including information theory, probability theory, dynamical systems and statistical physics, with applications in electrical engineering, computer science and molecular biology. This collection of research and survey papers presents important new results and open problems, serving as a unifying gateway for researchers in these areas. Based on talks given at the Banff International Research Station Workshop, 2007, this volume addresses a central problem of the subject: computation of the Shannon entropy rate of an HMP. This is a key quantity in statistical physics and information theory, characterising the fundamental limit on compression and closely related to channel capacity, the limit on reliable communication. Also discussed, from a symbolic dynamics and thermodynamical viewpoint, is the problem of characterizing the mappings between dynamical systems which map Markov measures to Markov (or Gibbs) measures, and which allow for Markov lifts of Markov chains.