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Binary entropy function — Wikipedia (en)

Source: Wikipedia (en)
Original: https://en.wikipedia.org/wiki/Binary_entropy_function
License (as archived): CC-BY-SA-4.0
Archived: 2026-10-31T09:44:01.757Z
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Binary entropy function — Wikipedia (en) — Wikipedia (en), https://en.wikipedia.org/wiki/Binary_entropy_function (CC-BY-SA-4.0); archived copy: https://metraipolis.com/documento/dc_67bc6ddceefb (Metraipolis archive dc_67bc6ddceefb, sha256 cb4aeadd8da6b6ef2bcc036835311385776f225098910cb9d9aed7a0f1e74318)

BibTeX

@misc{dc67bc6ddceefb,
  title = {Binary entropy function — Wikipedia (en)},
  howpublished = {Wikipedia (en)},
  url = {https://en.wikipedia.org/wiki/Binary_entropy_function},
  note = {License CC-BY-SA-4.0; archived copy https://metraipolis.com/documento/dc_67bc6ddceefb; sha256 cb4aeadd8da6b6ef2bcc036835311385776f225098910cb9d9aed7a0f1e74318},
  year = {2026}
}

Archived text

Reproduced under CC-BY-SA-4.0. Attribution: Wikipedia (en), https://en.wikipedia.org/wiki/Binary_entropy_function. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

In information theory, the binary entropy function, denoted H ( p ) {\displaystyle \operatorname {H} (p)} or H b ( p ) {\displaystyle \operatorname {H} _{\text{b}}(p)} , is defined as the entropy of a Bernoulli process (i.i.d. binary variable) X {\displaystyle X} with probability p {\displaystyle p} of one of two values, and is given by the formula: H ( X ) = − p log p − ( 1 − p ) log ( 1 − p ) . {\displaystyle \operatorname {H} (X)=-p\log p-(1-p)\log(1-p).} The base of the logarithm corresponds to the choice of units of information; base e (natural logarithm) corresponds to nats and is mathematically convenient, while base 2 (binary logarithm) corresponds to shannons and is conventional (as shown in the graph); explicitly: H ( X ) = − p log 2 p − ( 1 − p ) log 2 ( 1 − p ) . {\displaystyle \operatorname {H} (X)=-p\log _{2}p-(1-p)\log _{2}(1-p).} Note that the values at 0 and 1 are given by the limit 0 log 0 := lim x → 0 + x log x = 0 {\displaystyle \textstyle 0\log 0:=\lim _{x\to 0^{+}}x\log x=0} (by L'Hôpital's rule); and that "binary" refers to two possible values for the variable, not the units of information. When p = 1 / 2 {\displaystyle p=1/2} , the binary entropy function attains its maximum value of 1 shannon (1 binary unit of information); this is the case of an unbiased coin flip. When p = 0 {\displaystyle p=0} or p = 1 {\displaystyle p=1} , the binary entropy is 0 (regardless of the unit), corresponding to no information, since there is no uncertainty in the variable.

Tema degli esploratori: matematica e informatica teorica (deposito sistematico; ricerca «information theory»).
Fonte: Wikipedia in inglese, voce «Binary entropy function» — https://en.wikipedia.org/wiki/Binary_entropy_function
Wikidata Q4913893: «entropy of a process with only two probable values» (dati Wikidata in CC0 1.0) — https://www.wikidata.org/wiki/Q4913893
Autori: i contributori di Wikipedia (cronologia: https://en.wikipedia.org/wiki/Binary_entropy_function?action=history).
Licenza: Creative Commons Attribuzione-Condividi allo stesso modo 4.0 (CC BY-SA 4.0) — https://creativecommons.org/licenses/by-sa/4.0/. Il riuso richiede attribuzione e stessa licenza.
Estratto dell'introduzione non modificato nel contenuto; revisione 1324263438, consultata il 2026-10-31T09:32:06.216Z tramite l'API ufficiale di Wikipedia. Impronta SHA-256 dell'estratto: a5582afb84c1e12afc95a4355a140b0a0b6f6af1e0b8c5413e00ea7f6af7f599