8,487 research outputs found
Entropy rate of continuous-state hidden Markov chains
We prove that under mild positivity assumptions, the entropy rate of a continuous-state hidden Markov chain, observed when passing a finite-state Markov chain through a discrete-time continuous-output channel, is analytic as a function of the transition probabilities of the underlying Markov chain. We further prove that the entropy rate of a continuous-state hidden Markov chain, observed when passing a mixing finite-type constrained Markov chain through a discrete-time Gaussian channel, is smooth as a function of the transition probabilities of the underlying Markov chain. © 2010 IEEE.published_or_final_versionThe IEEE International Symposium on Information Theory (ISIT 2010), Austin, TX., 13-18 June 2010. In Proceedings of ISIT, 2010, p. 1468-147
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
Entropy rate calculations of algebraic measures
Let . We use a special class of translation invariant
measures on called algebraic measures to study the entropy rate
of a hidden Markov processes. Under some irreducibility assumptions of the
Markov transition matrix we derive exact formulas for the entropy rate of a
general state hidden Markov process derived from a Markov source corrupted
by a specific noise model. We obtain upper bounds on the error when using an
approximation to the formulas and numerically compute the entropy rates of two
and three state hidden Markov models
Consistency of the maximum likelihood estimator for general hidden Markov models
Consider a parametrized family of general hidden Markov models, where both
the observed and unobserved components take values in a complete separable
metric space. We prove that the maximum likelihood estimator (MLE) of the
parameter is strongly consistent under a rather minimal set of assumptions. As
special cases of our main result, we obtain consistency in a large class of
nonlinear state space models, as well as general results on linear Gaussian
state space models and finite state models. A novel aspect of our approach is
an information-theoretic technique for proving identifiability, which does not
require an explicit representation for the relative entropy rate. Our method of
proof could therefore form a foundation for the investigation of MLE
consistency in more general dependent and non-Markovian time series. Also of
independent interest is a general concentration inequality for -uniformly
ergodic Markov chains.Comment: Published in at http://dx.doi.org/10.1214/10-AOS834 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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
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