I've been looking up the problem of deciding appropriate binwidths for histograms and here's my broad-level understanding so far:
If we have $n$ data points, we assume that they're realizations of $n$ random variables following an unknown distribution $f$. Histograms are essentially density estimators that attempt to derive an estimate $\hat{f}$ of the underlying distribution. $\hat{f}$ depends on our choice of binwidth $h$, and an appropriate choice would give us an estimate that is close to the original distribution. The "closeness" is characterized by integrated mean square error.
There are a few rules for binning as given on Wikipedia, like the Freedman-Diaconis' choice, Sturges' rule and so on. My question is: how were these rules derived in the first place given that we don't know $f$? If we don't know $f$, we can't explicitly calculate IMSE and no optimization can be done. Were these rules applied to simulated data sets generated from a wide variety of probability distributions, and selected because they worked in most of those cases?
I'm not looking for exact derivations at this point, just the paradigm.
