We prove that under certain mild assumptions, the entropy rate of a hidden Markov chain, observed when passing a finite-state stationary Markov chain through a discrete-time continuous-output channel, is jointly analytic as a function of the input Markov chain parameters and the channel parameters. In particular, as consequences of the main theorems, we obtain analyticity for the entropy rate associated with representative channels: Cauchy and Gaussian.
We study the entropy rate of a hidden Markov process (HMP) defined by observing the output of a binary symmetric channel whose input is a first-order binary Markov process. Despite the simplicity of the models involved, the characterization of this entropy is a long standing open problem. By presenting the probability of a sequence under the model as a product of random matrices, one can see that the entropy rate sought is equal to a top Lyapunov exponent of the product. This offers an explanation for the elusiveness of explicit expressions for the HMP entropy rate, as Lyapunov exponents are notoriously difficult to compute. Consequently, we focus on asymptotic estimates, and apply the same product of random matrices to derive an explicit expression for a Taylor approximation of the entropy rate with respect to the parameter of the binary symmetric channel. The accuracy of the approximation is validated against empirical simulation results. We also extend our results to higher-order Markov processes and to Renyi entropies of any order. 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 [28], 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 flnite 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) 4 P(X = x;Y = y), x 2 X;y 2 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), respectively, when it is clear from the context). The entropy H(X) of the discrete random variable X, which measures the level of uncertainty of X, is deflned as (in this paper log is taken to mean the natural logarithm)  A recent result presented the expansion for the entropy rate of a hidden Markov process (HMP) as a power series in the noise variable epsi. The coefficients of the expansion around the noiseless (epsi=0) limit 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