**ETA:**This is in the literature, without justification as far as I can tell. See below.

I've worked out the details as best I can in an ipython notebook. It's a pain to include ipython notebooks directly into blogs at this point, so I've posted it on the ipython notebook viewer. I'll just add a few comments here.

The math works out, at least based on what I think is a reasonable mathematical model for the problem. I also ran statistical tests based on the same model, and they confirm that it works well enough for reasonably large numbers of photons, provided that the weight isn't dominated by a small number of heavily-weighted photons. And the formula is convenient and easy to remember.

**Edited to add**: I found a paper, Espinoza et al. 2012, that uses this formula, and it cites Pletsch et al. 2012, which simply states that this is the right formula. There's also Guillemot et al. 2012, which uses a Monte Carlo approach - essentially using all the bins as a population from which to draw the photon weights. But since Lucas Guillemot is the second author on the Espinoza et al. paper, I'm guessing they've figured out that the simple formula is the way to go.

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