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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
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 11 (7 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
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.
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 5 (4 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
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
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 3 (3 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
Analyticity of Entropy Rate of ContinuousState Hidden Markov Chains
, 2014
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Asymptotics of Noisy Constrained Channel Capacity
, 2007
"... In this paper, we generalize a result in [17] and derive an asymptotic formula for the entropy rate of a hidden Markov chain, observed when a Markov chain passes through a binary symmetric channel. And we prove an asymptotic formula for the capacity of a binary symmetric channel with input process s ..."
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In this paper, we generalize a result in [17] and derive an asymptotic formula for the entropy rate of a hidden Markov chain, observed when a Markov chain passes through a binary symmetric channel. And we prove an asymptotic formula for the capacity of a binary symmetric channel with input process supported on an irreducible finite type constraint. 1
THE ENTROPY RATE OF A BINARY CHANNEL WITH SLOWLY SWITCHING INPUT
, 2006
"... Abstract. In this note an asymptotic lower bound is derived for the entropy rate of the output of binary channel, whose input is a slowly switching Markov chain. The proof relies on certain concentration properties of conditional distribution (filtering) process. 1. ..."
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Abstract. In this note an asymptotic lower bound is derived for the entropy rate of the output of binary channel, whose input is a slowly switching Markov chain. The proof relies on certain concentration properties of conditional distribution (filtering) process. 1.