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Oh just to quickly reply to my old ponderings... I was slightly mixed up before. To get the scaling function you'd want to take the integral of the *inverse* of that bin width function.

Also, there's an easier procedure than that mess I was describing before:

Just do a regression to fit the data points of "bin number" versus "bin center" (and possibly add non-center points with the same bin number too), while ensuring the function resulting from regression is monotonic. That'll give the final scaling function straight out of the regression without needing any other funny business.