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Given a probability vector $v=(v_1,\ldots,v_n)$ with $1\geq v_i\geq 0$ and $\sum_{i=1}^n v_i=1$ its entropy can be defined as: $$ H(v):=-\sum_{i=1}^nv_i\log v_i $$ I wonder what is known about transformations that leave the entropy of the vector unchanged. That is, given $v$, a map $T:\mathbb R^n \to \mathbb R^n$ that takes probability vectors to probability vectors and verifies $H(v)=H(T(v))$.

A permutation would be a trivial one. For extremal vector $(0,\ldots ,1,\ldots,0)$ this would be the only solution. However, in general, there is an infinity of valid transformations. Is anything known about the structure of these maps?

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    $\begingroup$ Not clear in general. For example there the vectors $(1/4,1/4,1/4,1/4)$ and $(1/2,1/8,1/8,1/8,1/8)$ give the same entropy, but even if you fix the size of your vector there are many that give the same entropy. I have not seen any result about such transformations $T$. $\endgroup$ – MHS Feb 13 '15 at 13:30
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    $\begingroup$ You can think of equi-entropy contours on the probability simplex. The shape of these are often shown in papers, but unfortunately, I don't have a reference on top of my head. EDIT: here's one books.google.com/… and arxiv.org/pdf/1109.6440.pdf $\endgroup$ – Memming Feb 16 '15 at 22:44
  • $\begingroup$ I understand that you can draw equientropic contours on the simplex, but I would expect (or more precisely like) points with equal entropy to have some underlying structure $\endgroup$ – Euclean Feb 17 '15 at 8:32
  • $\begingroup$ @Memming no answer so far and the bounty expires in a couple of days. If you want to expand your comment into an answer I would accept it (provide it no other answer appears) $\endgroup$ – Euclean Feb 20 '15 at 15:01
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Here is a visualization of the isoentropy contours on 3 variable simplex (from Information Theoretic Learning: Renyi’s Entropy and Kernel Perspectives. Disclaimer: I contributed to a chapter in this book).

enter image description here

You could find what local changes can be applied by studying the tangent of the manifold perhaps.

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