On Jun 21, 2013, at 2:52 PM, James Holton <[email protected]> wrote: > Yes, but the DIFFERENCE between two Poisson-distributed values can be > negative. This is, unfortunately, what you get when you subtract the > background out from under a spot. Perhaps this is the source of confusion > here?
Maybe, but if you assume Poisson background and intensities, the ML estimate when background > measured intensity is not negative, nor is it the difference Ispot-Iback. The ML estimate is 0. (With a finite non-zero SD, smaller SD the smaller the Ispot/Iback ratio). > On Fri, Jun 21, 2013 at 11:34 AM, Douglas Theobald <[email protected]> > wrote: > I kinda think we're saying the same thing, sort of. > > You don't like the Gaussian assumption, and neither do I. If you make the > reasonable Poisson assumptions, then you don't get the Ispot-Iback=Iobs for > the best estimate of Itrue. Except as an approximation for large values, but > we are talking about the case when Iback>Ispot, where the Gaussian > approximation to the Poisson no longer holds. The sum of two Poisson > variates is also Poisson, which also can never be negative, unlike the > Gaussian. > > So I reiterate: the Ispot-Iback=Iobs equation assumes Gaussians and hence > negativity. The Ispot-Iback=Iobs does not follow from a Poisson assumption. > > > On Jun 21, 2013, at 1:13 PM, Ian Tickle <[email protected]> wrote: > > > On 21 June 2013 17:10, Douglas Theobald <[email protected]> wrote: > >> Yes there is. The only way you can get a negative estimate is to make > >> unphysical assumptions. Namely, the estimate Ispot-Iback=Iobs assumes > >> that both the true value of I and the background noise come from a > >> Gaussian distribution that is allowed to have negative values. Both of > >> those assumptions are unphysical. > > > > Actually that's not correct: Ispot and Iback are both assumed to come from > > a _Poisson_ distribution which by definition is zero for negative values of > > its argument (you can't have a negative number of photons), so are _not_ > > allowed to have negative values. For large values of the argument (in fact > > the approximation is pretty good even for x ~ 10) a Poisson approximates to > > a Gaussian, and then of course the difference Ispot-Iback is also > > approximately Gaussian. > > > > But I think that doesn't affect your argument. > > > > Cheers > > > > -- Ian >
