Information source (mathematics)

In mathematics, an information source is a sequence of random variables ranging over a finite alphabet Γ, having a stationary distribution.

The uncertainty, or entropy rate, of an information source is defined as

${\displaystyle H\{\mathbf {X} \}=\lim _{n\to \infty }H(X_{n}|X_{0},X_{1},\dots ,X_{n-1})}$

where

${\displaystyle X_{0},X_{1},\dots ,X_{n}}$

is the sequence of random variables defining the information source, and

${\displaystyle H(X_{n}|X_{0},X_{1},\dots ,X_{n-1})}$

is the conditional information entropy of the sequence of random variables. Equivalently, one has

${\displaystyle H\{\mathbf {X} \}=\lim _{n\to \infty }{\frac {H(X_{0},X_{1},\dots ,X_{n-1},X_{n})}{n+1}}.}$