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[[Image:Binary entropy plot.svg|thumbnail|right|200px|Entropy of a [[Bernoulli trial]] as a function of success probability, called the '''binary entropy function'''.]]
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In [[information theory]], the '''binary entropy function''', denoted <math>H(p) \,</math> or <math>H_{\mathrm b}(p) \,</math>, is defined as the [[information entropy|entropy]] of a [[Bernoulli process]] with [[probability]] of success ''p''.   Mathematically, the Bernoulli trial is modelled as a [[random variable]] ''X'' that can take on only two values: 0 and 1. The event <math>X = 1</math> is considered a success and the event <math>X = 0</math> is considered a failure. (These two events are mutually exclusive and exhaustive.)
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If <math> \mathrm{Pr}(X=1) = p,</math> then <math> \mathrm{Pr}(X=0) = 1-p </math> and the entropy of ''X'' is given by
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:<math>H(X) = H_{\mathrm b}(p) = -p \log_2 p - (1 - p) \log_2 (1 - p). \,</math>
 
where <math>0 \log_2 0</math> is taken to be 0.  The logarithms in this formula are usually taken (as shown in the graph) to the base 2. See ''[[binary logarithm]]''.
 
When <math>p=\frac 1 2 ,</math> the binary entropy function attains its maximum value. This is the case of the unbiased [[bit]], the most common unit of [[information entropy]].
 
<math>H(p)</math> is distinguished from the [[Information entropy|entropy function]] <math>H(X)</math> in that the former takes a single real number as a [[parameter]] whereas the latter takes a distribution or random variables as a parameter.
Sometimes the binary entropy function is also written as <math>H_2(p)</math>.
However, it is different from and should not be confused with the [[Rényi entropy]], which is denoted as <math>H_2(X)</math>.
 
==Explanation==
 
In terms of information theory, ''entropy'' is considered to be a measure of the uncertainty in a message. To put it intuitively, suppose <math>p=0</math>. At this probability, the event is certain never to occur, and so there is no uncertainty at all, leading to an entropy of 0. If <math>p=1</math>, the result is again certain, so the entropy is 0 here as well. When <math>p=1/2</math>, the uncertainty is at a maximum; if one were to place a fair bet on the outcome in this case, there is no advantage to be gained with prior knowledge of the probabilities. In this case, the entropy is maximum at a value of 1 bit. Intermediate values fall between these cases; for instance, if <math>p=1/4</math>, there is still a measure of uncertainty on the outcome, but one can still predict the outcome correctly more often than not, so the uncertainty measure, or entropy, is less than 1 full bit.
 
==Derivative==
The [[derivative]] of the '''binary entropy function''' may be expressed as the negative of the [[logit]] function:
:<math> {d \over dp} H_{\mathrm b}(p) = - \operatorname{logit}_2(p) = -\log_2\left( \frac{p}{1-p} \right). \,</math>
 
==Taylor series==
The [[Taylor series]] of the binary entropy function in a neighborhood of 1/2 is
:<math>H_{\mathrm b}(p) = 1 - \frac{1}{2\ln 2} \sum^{\infin}_{n=1} \frac{(1-2p)^{2n}}{n(2n-1)} </math>
for <math>0\le p\le 1</math>.
 
==See also==
* [[Metric entropy]]
* [[Information theory]]
* [[Information entropy]]
 
==References==
* David J. C. MacKay. ''[http://www.inference.phy.cam.ac.uk/mackay/itila/book.html Information Theory, Inference, and Learning Algorithms]'' Cambridge: Cambridge University Press, 2003. ISBN 0-521-64298-1
 
==External links==
 
 
[[Category:Entropy and information]]

Latest revision as of 04:08, 21 February 2014

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