Hi, The maximum likelihood estimator for a Poisson distributed variable is equal to the mean of the observations. In the case of a single observation, it will be equal to that observation. As Graeme suggested, you can calculate the probability mass function for a given observation with different Poisson parameters (i.e. true means) and see that function peaks when the parameter matches the observation.
The root mean squared error of the estimation of the true mean from a single observation k seems to be sqrt(k+2). Or to put it in another way, mean squared error, that is the expected value of (k-u)**2, for an observation k and a true mean u, is equal to k+2. You can see some example calculations at https://colab.research.google.com/drive/1eoaNrDqaPnP-4FTGiNZxMllP7SFHkQuS?usp=sharing Cheers, Filipe On Wed, 13 Oct 2021 at 17:14, Winter, Graeme (DLSLtd,RAL,LSCI) < [email protected]> wrote: > This rang a bell to me last night, and I think you can derive this from > first principles > > If you assume an observation of N counts, you can calculate the > probability of such an observation for a given Poisson rate constant X. If > you then integrate over all possible value of X to work out the central > value of the rate constant which is most likely to result in an observation > of N I think you get X = N+1 > > I think it is the kind of calculation you can perform on a napkin, if > memory serves > > All the best Graeme > > On 13 Oct 2021, at 16:10, Andrew Leslie - MRC LMB < > [email protected]> wrote: > > Hi Ian, James, > > I have a strong feeling that I have seen this result > before, and it was due to Andy Hammersley at ESRF. I’ve done a literature > search and there is a paper relating to errors in analysis of counting > statistics (se below), but I had a quick look at this and could not find > the (N+1) correction, so it must have been somewhere else. I Have cc’d Andy > on this Email (hoping that this Email address from 2016 still works) and > maybe he can throw more light on this. What I remember at the time I saw > this was the simplicity of the correction. > > Cheers, > > Andrew > > Reducing bias in the analysis of counting statistics data > Hammersley, AP <https://www.webofscience.com/wos/author/record/2665675> > (Hammersley, > AP) Antoniadis, A > <https://www.webofscience.com/wos/author/record/13070551> (Antoniadis, A) > NUCLEAR INSTRUMENTS & METHODS IN PHYSICS RESEARCH SECTION A-ACCELERATORS > SPECTROMETERS DETECTORS AND ASSOCIATED EQUIPMENT Volume 394 > Issue 1-2 > Page 219-224 > DOI 10.1016/S0168-9002(97)00668-2 > Published JUL 11 1997 > > On 12 Oct 2021, at 18:55, Ian Tickle <[email protected]> wrote: > > > Hi James > > What the Poisson distribution tells you is that if the true count is N > then the expectation and variance are also N. That's not the same thing as > saying that for an observed count N the expectation and variance are N. > Consider all those cases where the observed count is exactly zero. That > can arise from any number of true counts, though as you noted larger values > become increasingly unlikely. However those true counts are all >= 0 which > means that the mean and variance of those true counts must be positive and > non-zero. From your results they are both 1 though I haven't been through > the algebra to prove it. > > So what you are saying seems correct: for N observed counts we should be > taking the best estimate of the true value and variance as N+1. For > reasonably large N the difference is small but if you are concerned with > weak images it might start to become significant. > > Cheers > > -- Ian > > > On Tue, 12 Oct 2021 at 17:56, James Holton <[email protected]> wrote: > >> All my life I have believed that if you're counting photons then the >> error of observing N counts is sqrt(N). However, a calculation I just >> performed suggests its actually sqrt(N+1). >> >> My purpose here is to understand the weak-image limit of data >> processing. Question is: for a given pixel, if one photon is all you >> got, what do you "know"? >> >> I simulated millions of 1-second experiments. For each I used a "true" >> beam intensity (Itrue) between 0.001 and 20 photons/s. That is, for >> Itrue= 0.001 the average over a very long exposure would be 1 photon >> every 1000 seconds or so. For a 1-second exposure the observed count (N) >> is almost always zero. About 1 in 1000 of them will see one photon, and >> roughly 1 in a million will get N=2. I do 10,000 such experiments and >> put the results into a pile. I then repeat with Itrue=0.002, >> Itrue=0.003, etc. All the way up to Itrue = 20. At Itrue > 20 I never >> see N=1, not even in 1e7 experiments. With Itrue=0, I also see no N=1 >> events. >> Now I go through my pile of results and extract those with N=1, and >> count up the number of times a given Itrue produced such an event. The >> histogram of Itrue values in this subset is itself Poisson, but with >> mean = 2 ! If I similarly count up events where 2 and only 2 photons >> were seen, the mean Itrue is 3. And if I look at only zero-count events >> the mean and standard deviation is unity. >> >> Does that mean the error of observing N counts is really sqrt(N+1) ? >> >> I admit that this little exercise assumes that the distribution of Itrue >> is uniform between 0.001 and 20, but given that one photon has been >> observed Itrue values outside this range are highly unlikely. The >> Itrue=0.001 and N=1 events are only a tiny fraction of the whole. So, I >> wold say that even if the prior distribution is not uniform, it is >> certainly bracketed. Now, Itrue=0 is possible if the shutter didn't >> open, but if the rest of the detector pixels have N=~1, doesn't this >> affect the prior distribution of Itrue on our pixel of interest? >> >> Of course, two or more photons are better than one, but these days with >> small crystals and big detectors N=1 is no longer a trivial situation. >> I look forward to hearing your take on this. And no, this is not a trick. >> >> -James Holton >> MAD Scientist >> >> ######################################################################## >> >> To unsubscribe from the CCP4BB list, click the following link: >> https://www.jiscmail.ac.uk/cgi-bin/WA-JISC.exe?SUBED1=CCP4BB&A=1 >> >> This message was issued to members of www.jiscmail.ac.uk/CCP4BB, a >> mailing list hosted by www.jiscmail.ac.uk, terms & conditions are >> available at https://www.jiscmail.ac.uk/policyandsecurity/ >> > > ------------------------------ > > To unsubscribe from the CCP4BB list, click the following link: > https://www.jiscmail.ac.uk/cgi-bin/WA-JISC.exe?SUBED1=CCP4BB&A=1 > > > > ------------------------------ > > To unsubscribe from the CCP4BB list, click the following link: > https://www.jiscmail.ac.uk/cgi-bin/WA-JISC.exe?SUBED1=CCP4BB&A=1 > > > > > -- > > This e-mail and any attachments may contain confidential, copyright and or > privileged material, and are for the use of the intended addressee only. 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Registered in England > and Wales with its registered office at Diamond House, Harwell Science and > Innovation Campus, Didcot, Oxfordshire, OX11 0DE, United Kingdom > > > ------------------------------ > > To unsubscribe from the CCP4BB list, click the following link: > https://www.jiscmail.ac.uk/cgi-bin/WA-JISC.exe?SUBED1=CCP4BB&A=1 > ######################################################################## To unsubscribe from the CCP4BB list, click the following link: https://www.jiscmail.ac.uk/cgi-bin/WA-JISC.exe?SUBED1=CCP4BB&A=1 This message was issued to members of www.jiscmail.ac.uk/CCP4BB, a mailing list hosted by www.jiscmail.ac.uk, terms & conditions are available at https://www.jiscmail.ac.uk/policyandsecurity/
