On Jun 21, 2013, at 2:52 PM, James Holton <[log in to unmask]> 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 <[log in to unmask]> 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 <[log in to unmask]> wrote:
>
> > On 21 June 2013 17:10, Douglas Theobald <[log in to unmask]> 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
>
|