Binary entropy function

In information theory, the binary entropy function, denoted $\operatorname {H} (p)$ or $\operatorname {H} _{\text{b}}(p)$ , is defined as the entropy of a Bernoulli process with probability $p$ of one of two values. It is a special case of $\mathrm {H} (X)$ , the entropy function. Mathematically, the Bernoulli trial is modelled as a random variable $X$ that can take on only two values: 0 and 1, which are mutually exclusive and exhaustive.

Entropy of a Bernoulli trial as a function of binary outcome probability, called the binary entropy function.

If $\operatorname {Pr} (X=1)=p$ , then $\operatorname {Pr} (X=0)=1-p$ and the entropy of $X$ (in shannons) is given by

\operatorname {H} (X)=\operatorname {H} _{\text{b}}(p)=-p\log _{2}p-(1-p)\log _{2}(1-p)

,

where $0\log _{2}0$ 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 $p={\tfrac {1}{2}}$ , the binary entropy function attains its maximum value. This is the case of an unbiased coin flip.

$\operatorname {H} (p)$ is distinguished from the entropy function $\mathrm {H} (X)$ in that the former takes a single real number as a parameter whereas the latter takes a distribution or random variable as a parameter. Sometimes the binary entropy function is also written as $\operatorname {H} _{2}(p)$ . However, it is different from and should not be confused with the Rényi entropy, which is denoted as $\mathrm {H} _{2}(X)$ .

Explanation

In terms of information theory, entropy is considered to be a measure of the uncertainty in a message. To put it intuitively, suppose $p=0$ . 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 $p=1$ , the result is again certain, so the entropy is 0 here as well. When $p=1/2$ , 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 $p=1/4$ , 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:

{d \over dp}\operatorname {H} _{\text{b}}(p)=-\operatorname {logit} _{2}(p)=-\log _{2}\left({\frac {p}{1-p}}\right)

.

Taylor series

The Taylor series of the binary entropy function in a neighborhood of 1/2 is

\operatorname {H} _{\text{b}}(p)=1-{\frac {1}{2\ln 2}}\sum _{n=1}^{\infty }{\frac {(1-2p)^{2n}}{n(2n-1)}}

for $0\leq p\leq 1$ .

Bounds

The following bounds hold for $0<p<1$ :[1]

\ln(2)\cdot \log _{2}(p)\cdot \log _{2}(1-p)\leq H_{\text{b}}(p)\leq \log _{2}(p)\cdot \log _{2}(1-p)

and

4p(1-p)\leq H_{\text{b}}(p)\leq (4p(1-p))^{(1/\ln 4)}

where $\ln$ denotes natural logarithm.

References

Topsøe, Flemming (2001). "Bounds for entropy and divergence for distributions over a two-element set". JIPAM. Journal of Inequalities in Pure & Applied Mathematics. 2 (2): Paper No. 25, 13 p.-Paper No. 25, 13 p.