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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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Cited by 19 (8 self)
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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
Derivatives of Entropy Rate in Special Families of Hidden Markov Chains
 Issue 7, July 2007, Page(s):2642
"... Consider a hidden Markov chain obtained as the observation process of an ordinary Markov chain corrupted by noise. Zuk, et. al. [13, 14] 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 derivative ..."
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Cited by 6 (2 self)
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Consider a hidden Markov chain obtained as the observation process of an ordinary Markov chain corrupted by noise. Zuk, et. al. [13, 14] 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 binary symmetric noise, and give an abstract formula for the first derivative in terms of a measure on the simplex due to Blackwell. 1
Asymptotics of the inputconstrained binary symmetric channel capacity
 Annals of Applied Probability
, 2009
"... 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 inputs are sequences chosen from a constrained set. Motivated by a result of Ordentlich and Weissman [In Proceedings of IEEE Information Theory Workshop ..."
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Cited by 6 (2 self)
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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 inputs are sequences chosen from a constrained set. Motivated by a result of Ordentlich and Weissman [In Proceedings of IEEE Information Theory Workshop (2004) 117–122], we derive an asymptotic formula (when the noise parameter is small) for the entropy rate of a hidden Markov chain, observed when a Markov chain passes through a BSC. Using this result, we establish an asymptotic formula for the capacity of a BSC with input process supported on an irreducible finite type constraint, as the noise parameter tends to zero. 1. Introduction and background. Let X,Y be discrete random variables with alphabet X,Y and joint probability mass function pX,Y (x,y) △ = P(X = x,Y = y), x ∈ X,y ∈ Y [for notational simplicity, we will write p(x,y) rather than pX,Y (x,y), similarly p(x),p(y) rather than pX(x),pY (y), resp., when it
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 2 (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
Asymptotics of Entropy Rate in Special Families of Hidden Markov Chains
, 2008
"... We generalize a result in [8] and derive an asymptotic formula for entropy rate of a hidden Markov chain around a “weak Black Hole”. We also discuss applications of the asymptotic formula to the asymptotic behaviors of certain channels. Index Terms–entropy, entropy rate, hidden Markov chain, hidden ..."
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Cited by 1 (1 self)
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We generalize a result in [8] and derive an asymptotic formula for entropy rate of a hidden Markov chain around a “weak Black Hole”. We also discuss applications of the asymptotic formula to the asymptotic behaviors of certain channels. Index Terms–entropy, entropy rate, hidden Markov chain, hidden Markov model, hidden Markov process 1
Concavity of Mutual Information Rate for InputRestricted FiniteState Memoryless Channels
"... Abstract—We consider a finitestate memoryless channel with i.i.d. channel state and the input Markov process supported on a mixing finitetype constraint. We discuss the asymptotic behavior of entropy rate of the output hidden Markov chain and deduce that the mutual information rate of such a chann ..."
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Cited by 1 (1 self)
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Abstract—We consider a finitestate memoryless channel with i.i.d. channel state and the input Markov process supported on a mixing finitetype constraint. We discuss the asymptotic behavior of entropy rate of the output hidden Markov chain and deduce that the mutual information rate of such a channel is concave with respect to the parameters of the input Markov processes at high signaltonoise ratio. In principle, the concavity result enables good numerical approximation of the maximum mutual information rate and capacity of such a channel. I. CHANNEL MODEL In this paper, we show that for certain inputrestricted finitestate memoryless channels, the mutual information rate, at high SNR, is effectively a concave function of Markov input processes of a given order. While not directly addressed here, the goal is to help estimate the maximum of this function and
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.
THE ENTROPY RATE OF THE BINARY SYMMETRIC CHANNEL IN THE RARE TRANSITIONS REGIME
, 2008
"... Abstract. Asymptotic bounds for the entropy rate of the output of a binary channel are derived, using certain concentration properties of the conditional source distribution. In the case of symmetric channel, the exact asymptotic formula is obtained. 1. ..."
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Abstract. Asymptotic bounds for the entropy rate of the output of a binary channel are derived, using certain concentration properties of the conditional source distribution. In the case of symmetric channel, the exact asymptotic formula is obtained. 1.