# Entropy power inequality

In information theory, the entropy power inequality is a result that relates to so-called "entropy power" of random variables. It shows that the entropy power of suitably well-behaved random variables is a superadditive function. The entropy power inequality was proved in 1948 by Claude Shannon in his seminal paper "A Mathematical Theory of Communication". Shannon also provided a sufficient condition for equality to hold; Stam (1959) showed that the condition is in fact necessary.

## Statement of the inequality

For a random variable X : Ω → Rn with probability density function f : Rn → R, the differential entropy of X, denoted h(X), is defined to be

${\displaystyle h(X)=-\int _{\mathbb {R} ^{n}}f(x)\log f(x)\,dx}$

and the entropy power of X, denoted N(X), is defined to be

${\displaystyle N(X)={\frac {1}{2\pi e}}e^{{\frac {2}{n}}h(X)}.}$

In particular, N(X) = |K| 1/n when X is normal distributed with covariance matrix K.

Let X and Y be independent random variables with probability density functions in the Lp space Lp(Rn) for some p > 1. Then

${\displaystyle N(X+Y)\geq N(X)+N(Y).\,}$

Moreover, equality holds if and only if X and Y are multivariate normal random variables with proportional covariance matrices.