# Entropy rate

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Entropy rate

The entropy rate of a stochastic process is, informally, the time density of the average information in a stochastic process. For stochastic processes with a countable index, the entropy rate $H\left(X\right)$ is the limit of the joint entropy of $n$ members of the process $X_k$ divided by $n$, as $n$ tends to infinity:

:$H\left(X\right) = lim_\left\{n o infty\right\} frac\left\{1\right\}\left\{n\right\} H\left(X_1, X_2, dots X_n\right)$

when the limit exists. An alternative, related quantity is:

:$H\text{'}\left(X\right) = lim_\left\{n o infty\right\} H\left(X_n|X_\left\{n-1\right\}, X_\left\{n-2\right\}, dots X_1\right)$

For strongly stationary stochastic processes, $H\left(X\right) = H\text{'}\left(X\right)$.

Entropy rates for Markov chains

Since a stochastic process defined by a Markov chain which is irreducible and aperiodic has a stationary distribution, the entropy rate is independent of the initial distribution.

For example, for such a Markov chain "Y""k" defined on a countable number of states, given the transition matrix "P""ij", "H"("Y") is given by:

:$displaystyle H\left(Y\right) = - sum_\left\{ij\right\} mu_i P_\left\{ij\right\} log P_\left\{ij\right\}$

where "&mu;""i" is the stationary distribution of the chain.

A simple consequence of this definition is that the entropy rate of an i.i.d. stochastic process has an entropy rate that is the same as the entropy of any individual member of the process.

ee also

* Information source (mathematics)
* Markov information source

References

* Cover, T. and Thomas, J. (1991) Elements of Information Theory, John Wiley and Sons, Inc., ISBN 0471062596 [http://www3.interscience.wiley.com/cgi-bin/bookhome/110438582?CRETRY=1&SRETRY=0]

* [http://www.eng.ox.ac.uk/samp Systems Analysis, Modelling and Prediction (SAMP), University of Oxford] MATLAB code for estimating information-theoretic quantities for stochastic processes.

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