21,750 research outputs found
The Entropy of a Binary Hidden Markov Process
The entropy of a binary symmetric Hidden Markov Process is calculated as an
expansion in the noise parameter epsilon. We map the problem onto a
one-dimensional Ising model in a large field of random signs and calculate the
expansion coefficients up to second order in epsilon. Using a conjecture we
extend the calculation to 11th order and discuss the convergence of the
resulting series
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
Novel Lower Bounds on the Entropy Rate of Binary Hidden Markov Processes
Recently, Samorodnitsky proved a strengthened version of Mrs. Gerber's Lemma,
where the output entropy of a binary symmetric channel is bounded in terms of
the average entropy of the input projected on a random subset of coordinates.
Here, this result is applied for deriving novel lower bounds on the entropy
rate of binary hidden Markov processes. For symmetric underlying Markov
processes, our bound improves upon the best known bound in the very noisy
regime. The nonsymmetric case is also considered, and explicit bounds are
derived for Markov processes that satisfy the -RLL constraint
Prediction and Generation of Binary Markov Processes: Can a Finite-State Fox Catch a Markov Mouse?
Understanding the generative mechanism of a natural system is a vital
component of the scientific method. Here, we investigate one of the fundamental
steps toward this goal by presenting the minimal generator of an arbitrary
binary Markov process. This is a class of processes whose predictive model is
well known. Surprisingly, the generative model requires three distinct
topologies for different regions of parameter space. We show that a previously
proposed generator for a particular set of binary Markov processes is, in fact,
not minimal. Our results shed the first quantitative light on the relative
(minimal) costs of prediction and generation. We find, for instance, that the
difference between prediction and generation is maximized when the process is
approximately independently, identically distributed.Comment: 12 pages, 12 figures;
http://csc.ucdavis.edu/~cmg/compmech/pubs/gmc.ht
Statistical Physics Analysis of Maximum a Posteriori Estimation for Multi-channel Hidden Markov Models
The performance of Maximum a posteriori (MAP) estimation is studied
analytically for binary symmetric multi-channel Hidden Markov processes. We
reduce the estimation problem to a 1D Ising spin model and define order
parameters that correspond to different characteristics of the MAP-estimated
sequence. The solution to the MAP estimation problem has different operational
regimes separated by first order phase transitions. The transition points for
-channel system with identical noise levels, are uniquely determined by
being odd or even, irrespective of the actual number of channels. We
demonstrate that for lower noise intensities, the number of solutions is
uniquely determined for odd , whereas for even there are exponentially
many solutions. We also develop a semi analytical approach to calculate the
estimation error without resorting to brute force simulations. Finally, we
examine the tradeoff between a system with single low-noise channel and one
with multiple noisy channels.Comment: The paper has been submitted to Journal of Statistical Physics with
submission number JOSS-S-12-0039
On Hidden Markov Processes with Infinite Excess Entropy
We investigate stationary hidden Markov processes for which mutual
information between the past and the future is infinite. It is assumed that the
number of observable states is finite and the number of hidden states is
countably infinite. Under this assumption, we show that the block mutual
information of a hidden Markov process is upper bounded by a power law
determined by the tail index of the hidden state distribution. Moreover, we
exhibit three examples of processes. The first example, considered previously,
is nonergodic and the mutual information between the blocks is bounded by the
logarithm of the block length. The second example is also nonergodic but the
mutual information between the blocks obeys a power law. The third example
obeys the power law and is ergodic.Comment: 12 page
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
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
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