Entropy Rates For Markov Chains
Since a stochastic process defined by a Markov chain that is irreducible and aperiodic has a stationary distribution, the entropy rate is independent of the initial distribution.
For example, for such a Markov chain Yk defined on a countable number of states, given the transition matrix Pij, H(Y) is given by:
where μi 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.
Read more about this topic: Entropy Rate
Famous quotes containing the words entropy, rates and/or chains:
“Just as the constant increase of entropy is the basic law of the universe, so it is the basic law of life to be ever more highly structured and to struggle against entropy.”
—Václav Havel (b. 1936)
“[The] elderly and timid single gentleman in Paris ... never drove down the Champs Elysees without expecting an accident, and commonly witnessing one; or found himself in the neighborhood of an official without calculating the chances of a bomb. So long as the rates of progress held good, these bombs would double in force and number every ten years.”
—Henry Brooks Adams (1838–1918)
“He that has his chains knocked off, and the prison doors set open to him, is perfectly at liberty, because he may either go or stay, as he best likes; though his preference be determined to stay, by the darkness of the night, or illness of the weather, or want of other lodging. He ceases not to be free, though the desire of some convenience to be had there absolutely determines his preference, and makes him stay in his prison.”
—John Locke (1632–1704)