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 ${\displaystyle H(X)}$ is the limit of the joint entropy of ${\displaystyle n}$ members of the process ${\displaystyle X_{k}}$ divided by ${\displaystyle n}$, as ${\displaystyle n}$ tends to infinity:

${\displaystyle H(X)=\lim _{n\to \infty }{\frac {1}{n}}H(X_{1},X_{2},\dots X_{n})}$

Information theory
Entropy Differential entropy Conditional entropy Joint entropy Mutual information Directed information Conditional mutual information Relative entropy Entropy rate Limiting density of discrete points
Asymptotic equipartition property Rate–distortion theory
Shannon's source coding theorem Channel capacity Noisy-channel coding theorem Shannon–Hartley theorem

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

${\displaystyle H'(X)=\lim _{n\to \infty }H(X_{n}|X_{n-1},X_{n-2},\dots X_{1})}$

For strongly stationary stochastic processes, ${\displaystyle H(X)=H'(X)}$. 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.[1]