Binary entropy function

In information theory, the binary entropy function, denoted $H(p)\,$ or $H_{\mathrm {b} }(p)\,$ , is defined as the entropy of a Bernoulli trial 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 $X=1$ is considered a success and the event $X=0$ is considered a failure. (These two events are mutually exclusive and exhaustive.)

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

H(X)=H_{\mathrm {b} }(p)=-p\log p-(1-p)\log(1-p).\,

where $0log0$ 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 $p={\frac {1}{2}},$ the binary entropy function attains its maximum value. This is the case of the unbiased bit, the most common unit of information entropy.

$H(p)$ is distinguished from the entropy function by its taking a single scalar constant parameter. For tutorial purposes, in which the reader may not distinguish the appropriate function by its argument, $H_{2}(p)$ is often used; however, this could confuse this function with the analogous function related to Rényi entropy, so $H_{\mbox{b}}(p)$ (with "b" not in italics) should be used to dispel ambiguity.