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Gibbs' inequality - Wikipedia

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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 𝑃 = { 𝑝 1 , … , 𝑝 𝑛 } and 𝑄 = { 𝑞 1 , … , 𝑞 𝑛 } are discrete probability distributions. Then with equality if and only if 𝑝 𝑖 = 𝑞 𝑖 for 𝑖 = 1 , … 𝑛 .[1]: 68  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:[2]: 34 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

Gibbs' inequality - Wikipedia Jump to content From Wikipedia, the free encyclopedia Statement in information theory Josiah Willard Gibbs 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 [ edit ] Suppose that P = { p 1 , … , p n } {\displaystyle P=\{p_{1},\ldots ,p_{n}\}} and Q = { q 1 , … , q

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