In mathematics, the entropy power inequality is a result in probability theory 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 information entropy of X, denoted h(X), is defined to be

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

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

$N(X)\; =\; frac1\{2\; pi\; e\}\; exp\; left(frac\{2\}\{n\}\; h(X)\; right).$

In particular,N(X) = |K| ^1/n when X ~ Φ_K.

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

$N(X\; +\; Y)\; geq\; N(X)\; +\; N(Y).\; ,$

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

References

• Dembo, Amir; Cover, Thomas M. and Thomas, Joy A. (1991). "Information-theoretic inequalities". IEEE Trans. Inform. Theory 37 (6): 1501–1518.
• Gardner, Richard J. (2002). "The Brunn-Minkowski inequality". Bull. Amer. Math. Soc. (N.S.) 39 (3): 355–405 (electronic).
• Shannon, Claude E. (1948). "A mathematical theory of communication". Bell System Tech. J. 27 379–423, 623–656.
• Stam, A.J. (1959). "Some inequalities satisfied by the quantities of information of Fisher and Shannon". Information and Control 2 101–112.