In information theory, Gibbs' inequality is a statement about the information 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.

Gibbs' inequality

Suppose that and are discrete probability distributions. Then with equality if and only if p_i = q_i for. 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.

Proof

For simplicity, we prove the statement using the natural logarithm, denoted by ln , since so the particular logarithm base b > 1 that we choose only scales the relationship by the factor 1 / ln b . Let I denote the set of all i for which pi is non-zero. Then, since for all x > 0, with equality if and only if x=1, we have: The last inequality is a consequence of the pi and qi being part of a probability distribution. Specifically, the sum of all non-zero values is 1. Some non-zero qi, however, may have been excluded since the choice of indices is conditioned upon the pi being non-zero. Therefore, the sum of the qi may be less than 1. So far, over the index set I, we have: or equivalently Both sums can be extended to all, i.e. including p_i=0, by recalling that the expression p \ln p tends to 0 as p tends to 0, and (-\ln q) tends to \infty as q tends to 0. We arrive at For equality to hold, we require This can happen if and only if p_i = q_i for.

Alternative proofs

The result can alternatively be proved using Jensen's inequality, the log sum inequality, or the fact that the Kullback-Leibler divergence is a form of Bregman divergence.

Proof by Jensen's inequality

Because log is a concave function, we have that: where the first inequality is due to Jensen's inequality, and q being a probability distribution implies the last equality. Furthermore, since \log is strictly concave, by the equality condition of Jensen's inequality we get equality when and Suppose that this ratio is \sigma, then we have that where we use the fact that p, q are probability distributions. Therefore, the equality happens when p = q.

Proof by Bregman divergence

Alternatively, it can be proved by noting thatfor all p, q > 0, with equality holding iff p=q. Then, sum over the states, we havewith equality holding iff p = q. This is because the KL divergence is the Bregman divergence generated by the function.

Corollary

The entropy of P is bounded by: The proof is trivial – simply set q_i = 1/n for all i.