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

enter image description here

Which produces a symmetric graph like this:

enter image description here

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)?

enter image description here

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.

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    $\begingroup$ 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 $p = 1 - 0.8 = 0.2$ $\endgroup$
    – KillaKem
    Commented Jul 5, 2021 at 11:21

1 Answer 1

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

enter image description here

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:

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

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

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