Entropy rate

In the mathematical theory of probability, the entropy rate or source information 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 formula_1 is the limit of the joint entropy of formula_2 members of the process formula_3 divided by formula_2, as formula_2 tends to infinity:
when the limit exists. An alternative, related quantity is:
For strongly stationary stochastic processes, formula_8. The entropy rate can be thought of as a general property of stochastic sources; this is the asymptotic equipartition property. The entropy rate may be used to estimate the complexity of stochastic processes. It is used in diverse applications ranging from characterizing the complexity of languages, blind source separation, through to optimizing quantizers and data compression algorithms. For example, a maximum entropy rate criterion may be used for feature selection in machine learning.
Entropy rates for Markov chains.
Since a stochastic process defined by a Markov chain that is irreducible, aperiodic
and positive recurrent has a stationary distribution, the entropy rate is independent of the initial distribution.
For example, for such a Markov chain formula_9 defined on a countable number of states, given the transition matrix formula_10, formula_11 is given by:
where formula_13 is the asymptotic distribution of the chain.
A simple consequence of this definition is that an i.i.d. stochastic process has an entropy rate that is the same as the entropy of any individual member of the process.