Gibbs' inequality

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. Suppose that is a discrete probability distribution. Then for any other probability distribution the following inequality between positive quantities (since pi and qi are between zero and 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. For simplicity, we prove the statement using the natural logarithm (ln). Because the particular logarithm base b > 1 that we choose only scales the relationship by the factor 1 / ln b. Let denote the set of all 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 , we have: or equivalently Both sums can be extended to all , i.e. including , by recalling that the expression tends to 0 as tends to 0, and tends to as tends to 0. We arrive at For equality to hold, we require for all so that the equality holds, and which means if , that is, if . This can happen if and only if for .