On 22 June 2013 18:04, Douglas Theobald <dtheob...@brandeis.edu> wrote:
> Ian, I really do think we are almost saying the same thing. Let me try to > clarify. > I agree, but still only "almost"! > --- but in truth the Poisson model does not account for other physical > sources of error that arise from real crystals and real detectors, such as > dark noise and read noise (that's why I would prefer a gamma distribution). > A photon counter is a digital device, not an analogue one. It starts at zero and adds 1 every time it detects a photon (or what it thinks is a photon). Once added, it is physically impossible for it to subtract 1 from its accumulated count: it contains no circuit to do that. It can certainly miss photons, so you end up with less than you should, and it can certainly 'see' photons where there were none (e.g. from instrumental noise), so you end up with more than you should. However once a count has been accumulated in the digital memory it stays there until the memory is cleared for the next measurement, and you can never end up with less than that accumulated count and in particular not less than zero; the bits of memory where the counts are accumulated are simply not programmed to return negative numbers. It has nothing to do with whether the crystal is real or not, all that matters is that photons from "somewhere" are arriving at and being counted by the detector. The accumulated counts at any moment in time have a Poisson distribution since the photons arrive completely randomly in time. > In the Poisson model you outline, Ispot is the sum of two Poisson > variables, Iback and Iobs. That means Ispot is also Poisson and can never > be negative. Again --- the observed data (Ispot) is a *sum*, so that is > what we must deal with. The likelihood function for this model is: > > No, Iobs is _not_ a Poisson variable, indeed I never said it was: I explained that it's the difference of 2 Poissonians Ispot and Iback and therefore approximately Gaussian (please re-read my previous email). So the sum of Poissonians does not come into it. The only Poissonian variates here are Ispot and Iback. Neither is the background under Ispot a Poissonian (let's call it Iback', so strictly speaking Ispot = Iobs + Iback' and Iback is an estimate of Iback', quite possibly with a non-random error). This is because Iobs and Iback' are not observable photon counts. QM does not allow you to separate Ispot into separate photon counts, because photons are indistinguishable. If the photons were labelled 'spot', 'back' and 'obs' then you could count Iobs independently and it would be a Poissonian (and that would indeed solve all our problems!). But, sadly, photons are indistinguishable, they don't arrive with handy labels! Does any of that change your view? Cheers -- Ian
