Is it practical to derive the prior distribution by dividing the posterior by the likelihood and multiplying by the "evidence"? Is it practical to derive the optimal prior distribution by dividing the posterior by the likelihood and multiplying by the "evidence"?
Suppose you assume a probability distribution.
You know what the posterior is because you assume you know what the asymptotic convergences are.
Now you can derive the prior by multiplication.

Mathematically, starting from given densities $\pi(\theta|x)$ and $f(x|\theta)$, there is no reason for these two functions to be compatible, namely for$$\dfrac{\pi(\theta|x)}{f(x|\theta)}$$to factorise as a function of $x$ and a function of $\theta$.
