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 L^p space L^p(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.

See also

• Information entropy
• Information theory
• Limiting density of discrete points
• Self-information
• Kullback–Leibler divergence
• Entropy estimation

References

• Dembo, Amir; Cover, Thomas M.; Thomas, Joy A. (1991). "Information-theoretic inequalities". IEEE Trans. Inf. Theory. 37 (6): 1501–1518. doi:10.1109/18.104312. MR 1134291. S2CID 845669.
• Gardner, Richard J. (2002). "The Brunn–Minkowski inequality". Bull. Amer. Math. Soc. (N.S.). 39 (3): 355–405 (electronic). doi:10.1090/S0273-0979-02-00941-2.
• Shannon, Claude E. (1948). "A mathematical theory of communication". Bell System Tech. J. 27 (3): 379–423, 623–656. doi:10.1002/j.1538-7305.1948.tb01338.x. hdl:10338.dmlcz/101429.
• Stam, A. J. (1959). "Some inequalities satisfied by the quantities of information of Fisher and Shannon". Information and Control. 2 (2): 101–112. doi:10.1016/S0019-9958(59)90348-1.