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Analyticity of entropy rate of a hidden Markov chain
 In Proc. of IEEE International Symposium on Information Theory, Adelaide, Australia, September 4September 9 2005
, 1995
"... We prove that under mild positivity assumptions the entropy rate of a hidden Markov chain varies analytically as a function of the underlying Markov chain parameters. A general principle to determine the domain of analyticity is stated. An example is given to estimate the radius of convergence for t ..."
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We prove that under mild positivity assumptions the entropy rate of a hidden Markov chain varies analytically as a function of the underlying Markov chain parameters. A general principle to determine the domain of analyticity is stated. An example is given to estimate the radius of convergence for the entropy rate. We then show that the positivity assumptions can be relaxed, and examples are given for the relaxed conditions. We study a special class of hidden Markov chains in more detail: binary hidden Markov chains with an unambiguous symbol, and we give necessary and sufficient conditions for analyticity of the entropy rate for this case. Finally, we show that under the positivity assumptions the hidden Markov chain itself varies analytically, in a strong sense, as a function of the underlying Markov chain parameters. 1
New bounds on the entropy rate of hidden Markov process
 IEEE Information Theory Workshop
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From FiniteSystem Entropy to Entropy Rate for a
 Hidden Markov Process. Signal Processing Letters, IEEE, Volume 13, Issue 9, Sept. 2006 Page(s):517
, 2006
"... Abstract—A recent result presented the expansion for the entropy rate of a hidden Markov process (HMP) as a power series in the noise variable. The coefficients of the expansion around the noiseless @ aHAlimit were calculated up to 11th order, using a conjecture that relates the entropy rate of an H ..."
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Abstract—A recent result presented the expansion for the entropy rate of a hidden Markov process (HMP) as a power series in the noise variable. The coefficients of the expansion around the noiseless @ aHAlimit were calculated up to 11th order, using a conjecture that relates the entropy rate of an HMP to the entropy of a process of finite length (which is calculated analytically). In this letter, we generalize and prove the conjecture and discuss its theoretical and practical consequences.
Derivatives of entropy rate in special families of hidden Markov chains
 IEEE Trans. Info. Theory
, 2007
"... Abstract—Consider a hidden Markov chain obtained as the observation process of an ordinary Markov chain corrupted by noise. Recently Zuk et al. showed how, in principle, one can explicitly compute the derivatives of the entropy rate of at extreme values of the noise. Namely, they showed that the der ..."
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Cited by 9 (4 self)
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Abstract—Consider a hidden Markov chain obtained as the observation process of an ordinary Markov chain corrupted by noise. Recently Zuk et al. showed how, in principle, one can explicitly compute the derivatives of the entropy rate of at extreme values of the noise. Namely, they showed that the derivatives of standard upper approximations to the entropy rate actually stabilize at an explicit finite time. We generalize this result to a natural class of hidden Markov chains called “Black Holes. ” We also discuss in depth special cases of binary Markov chains observed in binarysymmetric noise, and give an abstract formula for the first derivative in terms of a measure on the simplex due to Blackwell. Index Terms—Analyticity, entropy, entropy rate, hidden Markov chain, hidden Markov model, hidden Markov process.
Analyticity of Entropy Rate in Families of Hidden Markov Chains
, 2008
"... We prove that under a mild positivity assumption the entropy rate of a hidden Markov chain varies analytically as a function of the underlying Markov chain parameters. We give examples to show how this can fail in some cases. And we study two natural special classes of hidden Markov chains in more d ..."
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We prove that under a mild positivity assumption the entropy rate of a hidden Markov chain varies analytically as a function of the underlying Markov chain parameters. We give examples to show how this can fail in some cases. And we study two natural special classes of hidden Markov chains in more detail: binary hidden Markov chains with an unambiguous symbol and binary Markov chains corrupted by binary symmetric noise. Finally, we show that under the positivity assumption the hidden Markov chain itself varies analytically, in a strong sense, as a function of the underlying Markov chain parameters.
THE THEORY OF TRACKABILITY AND ROBUSTNESS FOR PROCESS DETECTION
, 2006
"... Many applications of current interests involve detecting instances of processes from databases or streams of sensor reports. Detecting processes relies on identifying evidences for the existence of such processes from usually noisy and incomplete observable events through statistical inferences. The ..."
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Cited by 4 (0 self)
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Many applications of current interests involve detecting instances of processes from databases or streams of sensor reports. Detecting processes relies on identifying evidences for the existence of such processes from usually noisy and incomplete observable events through statistical inferences. The performance of inferences can vary dramatically, depending on the complexity of processes ’ behavioral patterns, sensor resolution and sampling rate, SNR, location and coverage, and so on. Stochastic models are mathematical representations of all these factors. In this dissertation, we intend to answer the following questions: Performance – How accurate are the inference results given the model? Trackability – What are the boundaries of the performance of inferences? Robustness – How sensitive is the performance of inferences to perturbations on input data or model parameters? Methodology – How can we improve the trackability and robustness of process detection? From the information theoretic point of view, we address the reason of errors in detection to the losses of source information during the sensing stage, measured as entropy in the Shannon sense. We propose a series of entropic measures of the trackability and robustness for a popular modeling technique – hidden Markov models (HMM). Our major contributions include: the theory of trackability; structural analysis of trackability for HMMs through its nonparametric counterpart – DFA/NFAs; an effective visualization method for analyzing the trackability for
Approximations for the Entropy Rate of a Hidden Markov Process
"... Abstract—Let {Xt} be a stationary finitealphabet Markov chain and {Zt} denote its noisy version when corrupted by a discrete memoryless channel. We present an approach to bounding the entropy rate of {Zt} by the construction and study of a related measurevalued Markov process. To illustrate its ef ..."
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Abstract—Let {Xt} be a stationary finitealphabet Markov chain and {Zt} denote its noisy version when corrupted by a discrete memoryless channel. We present an approach to bounding the entropy rate of {Zt} by the construction and study of a related measurevalued Markov process. To illustrate its efficacy, we specialize it to the case of a BSCcorrupted binary Markov chain. The bounds obtained are sufficiently tight to characterize the behavior of the entropy rate in asymptotic regimes that exhibit a “concentration of the support”. Examples include the ‘high SNR’, ‘low SNR’, ‘rare spikes’, and ‘weak dependence’ regimes. Our analysis also gives rise to a deterministic algorithm for approximating the entropy rate, achieving the best known precisioncomplexity tradeoff, for a significant subset of the process parameter space. I.
Noisy Constrained Capacity for BSC Channels
"... Abstract — We study the classical problem of noisy constrained capacity in the case of the binary symmetric channel (BSC), namely, the capacity of a BSC whose input is a sequence from a constrained set. As stated in [4] “... while calculation of the noisefree capacity of constrained sequences is we ..."
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Abstract — We study the classical problem of noisy constrained capacity in the case of the binary symmetric channel (BSC), namely, the capacity of a BSC whose input is a sequence from a constrained set. As stated in [4] “... while calculation of the noisefree capacity of constrained sequences is well known, the computation of the capacity of a constraint in the presence of noise... has been an unsolved problem in the halfcentury since Shannon’s landmark paper.... ” We first express the constrained capacity of a binary symmetric channel with (d, k)constrained input as a limit of the top Lyapunov exponents of certain matrix random processes. Then, we compute asymptotic approximations of the noisy constrained capacity for cases where the noise parameter ε is small. In particular, we show that when k≤2d, the error term with respect to the constraint capacity is O(ε), whereas it is O(ε log ε) when k> 2d. In both cases, we compute the coefficient of the error term. In the course of establishing these findings, we also extend our previous results on the entropy of a hidden Markov process to higherorder finite memory processes. These conclusions are proved by a combination of analytic and combinatorial methods. I.
Entropy of Hidden Markov Processes and Connections to Dynamical Systems
, 2007
"... 1 Workshop Overview The focus of this workshop was entropy rate of Hidden Markov Processes (HMP)’s, related informationtheoretic quantities and other connections with related subjects. The workshop brought together thirty mathematicians, computer scientists and electrical engineers from institution ..."
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1 Workshop Overview The focus of this workshop was entropy rate of Hidden Markov Processes (HMP)’s, related informationtheoretic quantities and other connections with related subjects. The workshop brought together thirty mathematicians, computer scientists and electrical engineers from institutions in Canada, US, Europe, Latin America and Asia. While some participants were from industrial research organizations, such as HewlittPackard Labs and Philips Research, most were from universities, representing all academic ranks, including three postdocs and four graduate students. The participants came from a wide variety of academic disciplines, including information theory (source, channel and constrained coding), dynamical systems (symbolic dynamics, iterated function systems and Lyapunov exponents), probability theory, and statistical mechanics. There were eighteen 50minute lectures from Monday through Friday (see section 4 and the Appendices). This left plenty of time for spirited, yet informal, interaction at breaks, over meals, at a Sunday evening reception and a Wednesday afternoon hike. In addition, there were two problem sessions, one on Tuesday evening to collect and formulate open problems and one on Thursday evening to discuss approaches and possible solutions.