5.4 Differential entropy

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In this module we consider differential entropy.

Consider the entropy of continuous random variables. Whereas the (normal) entropy is the entropy of a discrete random variable, the differential entropy is the entropy of a continuous random variable.

Differential entropy

Differential entropy
The differential entropy $h(X)$ of a continuous random variable $X$ with a pdf $f(x)$ is defined as
$h(X)=-\int_{()} \,d x$ f x f x
Usually the logarithm is taken to be base 2, so that the unit of the differential entropy is bits/symbol. Note that is the discrete case, $h(X)$ depends only on the pdf of $X$ . Finally, we note that the differential entropy is the expected value of $-\lg f(x)$ , i.e.,
$h(X)=-E(\lg f(x))$

Now, consider a calculating the differential entropy of some random variables.

Consider a uniformly distributed random variable $X$ from $c$ to $c+\Delta$ . Then its density is $\frac{1}{\Delta }$ from $c$ to $c+\Delta$ , and zero otherwise.

We can then find its differential entropy as follows,

$h(X)=-\int_{c}^{c+\Delta } \frac{1}{\Delta }\lg \left(\frac{1}{\Delta }\right)\,d x=\lg \Delta$
Note that by making $\Delta$ arbitrarily small, the differential entropy can be made arbitrarily negative, while taking $\Delta$ arbitrarily large, the differential entropy becomes arbitrarily positive.

Consider a normal distributed random variable $X$ , with mean $m$ and variance $\sigma ^{2}$ . Then its density is $\sqrt{\frac{1}{2\pi \sigma ^{2}}}e^{-\left(\frac{(x-m)^{2}}{2\sigma ^{2}}\right)}$ .

We can then find its differential entropy as follows, first calculate $-\lg f(x)$ :

$-\lg f(x)=\frac{1}{2}\lg (2\pi \sigma ^{2})+\lg e()\frac{(x-m)^{2}}{2\sigma ^{2}}$
Then since $E((X-m)^{2})=\sigma ^{2}$ , we have
$h(X)=\frac{1}{2}\lg (2\pi \sigma ^{2})+\frac{1}{2}\lg e=\frac{1}{2}\lg (2\pi \times e\sigma ^{2})$

Properties of the differential entropy

In the section we list some properties of the differential entropy.

• The differential entropy can be negative
• $h(X+c)=h(X)$ , that is translation does not change the differential entropy.
• $h(aX)=h(X)+\lg \left|a\right|$ , that is scaling does change the differential entropy.
The first property is seen from both and . The two latter can be shown by using .

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