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I apologize for the very specific question I am asking. Define the relative entropy $D:[0,1]\times[0,1]\mapsto[0,\infty]$ by

$$D(x,y) = x\log_e\frac{x}{y}+(1-x)\log_e\frac{1-x}{1-y}.$$

Note $D(x,y)\geq 0$ with equality iff $x=y.$

It is possible to show that there exists a continuous decreasing bijection $f:[0,1]\mapsto [0,1]$ such that for $r=f(s),$ we have the equality

$$\frac{D(0.2+0.1r,0.2+0.1s)}{D(r,s)} = \frac{D(0.2+0.1s,0.2+0.1r)}{D(s,r)},$$

for all $s\in (0,1)$ except $s=s_0$ which satisfies $f(s_0)=s_0.$ [The constants 0.2,0.1 chosen here are generic. I am interested in the general case with two arbitrary constants a,b.]

One can observe that $f$ is an involution.

I would like to know

(a) if there is a systematic way to guess an explicit formula for a general involution;

(b) if there is a software package that can generate an explicit formula (when possible and not too hard) from an implicit one.

Thank you very much for your time.

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  • $\begingroup$ Why do you choose the constants $0.1$ and $0.2$ in your question? $\endgroup$
    – Stefan Kohl ♦
    Apr 28, 2014 at 8:50
  • $\begingroup$ Thanks Stefan. I edited the question to mention that the constants are generic and were chosen to be 0.1,0.2 to illustrate a concrete example. $\endgroup$
    – user50125
    Apr 28, 2014 at 9:13

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