# How do you create a skewed Shannon/Information Entropy?

The definition of Shannon Entropy for a Random variable with two outcomes is ( From https://en.wikipedia.org/wiki/Entropy_(information_theory)):

Which produces a symmetric graph like this:

However, I have use case where the uncertainty is treated as importance but I want to skew it so that higher probability has more importance. Basically, what formula can I use to skew the Entropy/Uncertainty either right or left?

Something which will produce a graph like this (similar to Skewed Normal Distribution)?

Note: I still want the Entropy range to be from 0-1. Can't use a weightage between p and 1-p since that doesn't make it 0-1.

• To me it doesn't make sense why you would want a graph like that, the y-axis is most definitely no longer a valid entropy measure. Remember a binary distribution with, for instance, $p = 0.8$ is exactly the same distribution as $p = 1 - 0.8 = 0.2$ Commented Jul 5, 2021 at 11:21

Here is a solution:

Consider the curves $$C_{a,b}$$ with parametric equations:

$$x(t)=t+abt(1-t), \ \ \ y(t)=at(1-t)$$

where $$0 \le a,b \le 1$$.

Fig. 1: Curves $$C_{a,b}$$ plotted for values $$a=0.6,0.8, 1$$ and $$b=0., 0.2, 0.4, 0.6, 0.8, 1$$. The red curve corresponds to the case $$a=1$$ and $$b=0.8$$.

Explanation: I have skewed the original curve by applying a "skewing matrix" to the original curve in this way:

$$\begin{pmatrix}1&1\\0&b\end{pmatrix}\begin{pmatrix}t\\at(1-t)\end{pmatrix}=\begin{pmatrix}t+at(1-t)\\abt(1-t)\end{pmatrix}$$

• Hi Jean, the image didn't come properly, so I have edited your answer ;D
– Babu
Commented Jul 5, 2021 at 9:13
• @Buraian Thanks ! Commented Jul 5, 2021 at 9:14
• Any comment ?... It is possible to obtain a cartesian representation $y=f(x)$ out of the parametric representation. Commented Jul 5, 2021 at 16:25