Gibbs' inequality

In information theory, Gibbs' inequality is a statement about the mathematical entropy of a discrete probability distribution. Several other bounds on the entropy of probability distributions are derived from Gibbs' inequality, including Fano's inequality. It was first presented by J. Willard Gibbs in the 19th century.

Suppose that

is a probability distribution. Then for any other probability distribution

the following inequality between positive quantities (since the p_i and q_i are positive numbers less than one) holds

with equality if and only if

for all i. Put in words, the information entropy of a distribution P is less than or equal to its cross entropy with any other distribution Q.

The difference between the two quantities is the Kullback–Leibler divergence or relative entropy, so the inequality can also be written:

Note that the use of base-2 logarithms is optional, and allows one to refer to the quantity on each side of the inequality as an "average surprisal" measured in bits.

Since

it is sufficient to prove the statement using the natural logarithm (ln). Note that the natural logarithm satisfies

for all x > 0 with equality if and only if x=1.

Let ${\displaystyle I}$ denote the set of all ${\displaystyle i}$ for which p_i is non-zero. Then

...
Wikipedia