4 research outputs found
Taylor series expansions for the entropy rate of Hidden Markov Processes
Finding the entropy rate of Hidden Markov Processes is an active research
topic, of both theoretical and practical importance. A recently used approach
is studying the asymptotic behavior of the entropy rate in various regimes. In
this paper we generalize and prove a previous conjecture relating the entropy
rate to entropies of finite systems. Building on our new theorems, we establish
series expansions for the entropy rate in two different regimes. We also study
the radius of convergence of the two series expansions
Derivatives of Entropy Rate in Special Families of Hidden Markov Chains
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.Comment: The relaxed condtions for entropy rate and examples are taken out (to
be part of another paper). The section about general principle and an example
to determine the domain of analyticity is taken out (to be part of another
paper). A section about binary Markov chains corrupted by binary symmetric
noise is adde
Analyticity of Entropy Rate of Hidden Markov Chains
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 {\em itself} varies
analytically, in a strong sense, as a function of the underlying Markov chain
parameters.Comment: The title has been changed. The new main theorem now combines the old
main theorem and the remark following the old main theorem. A new section is
added as an introduction to complex analysis. General principle and an
example to determine the domain of analyticity of entropy rate have been
added. Relaxed conditions for analyticity of entropy rate and the
corresponding examples are added. The section about binary markov chain
corrupted by binary symmetric noise is taken out (to be part of another
paper
A Randomized Algorithm for the Capacity of Finite-State Channels
Inspired by ideas from the field of stochastic approximation, we propose a ran- domized algorithm to compute the capacity of a finite-state channel with a Markovian input. When the mutual information rate of the channel is concave with respect to the chosen parameterization, the proposed algorithm proves to be convergent to the ca- pacity of the channel almost surely with the derived convergence rate. We also discuss the convergence behavior of the algorithm without the concavity assumption.published_or_final_versio