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I did a thing:
Surprisingly, the resulting uncertainties are bad while the fit looks
quite good. I've read that this sort of analysis gives a better estimation
of the true uncertainty than one often gets with least squares fitting.
Plotting a selection of 1000 models generated from this analysis shows that
they all lie very close to the mean. The assumptions of this model do not
constrain the model parameters given the particular data collected.
Said again: any of the terms in this model can be adjusted about 10%. You
can still get a similar and reasonable fit. You just need the right
tweaking of the other parameters.
Bad model assumptions here almost certainly include:
A) The background rate is constant
B) The stability of the detector