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

Abstract — Let {Xt} be a stationary finite-alphabet Markov chain and {Zt} denote its noisy version when corrupted by a discrete memoryless channel. Let P (Xt ∈ ·|Z t −∞) denote the conditional distribution of Xt given all past and present noisy observations, a simplex-valued random variable. We present a new approach to bounding the entropy rate of {Zt} by approximating the distribution of this random variable. This approximation is facilitated by the construction and study of a Markov process whose stationary distribution determines the distribution of P (Xt ∈ ·|Z t −∞). To illustrate the efficacy of this approach, we specialize it and derive concrete bounds for the case of a binary Markov chain corrupted by a binary symmetric channel (BSC). These bounds are seen to capture the behavior of the entropy rate in various asymptotic regimes. I.

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.

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. 1

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 noise-free 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 half-century since Shannon’s landmark paper.... ” We 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. 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

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 non-parametric counterpart – DFA/NFAs; an effective visualization method for analyzing the trackability for

Consider a hidden Markov chain obtained as the observation process of an ordinary Markov chain corrupted by noise. Zuk, et. al. [16, 17] 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

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 noise-free 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 half-century 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 higher-order finite memory processes. These conclusions are proved by a combination of analytic and combinatorial methods. I.