Assume we have a variable $x \in [x_{min},x_{max}]$, and an increasing function $f(x)$. And we want to estimate $x$ according to the sampling of $y=f(x)+n$, where $n$ is the Guassian $N(0,1)$.
Assume that we are constrained to sample at most $N$ points, and our target is to have a best estimation of $x$. In addition, we assume the prior distribution $x$ is uniform.
My question is that which sampling strategy is the best? In my opinion, there are two candidates, one is uniform over $x$ and the other is uniform over $f(x)$.