Dear Gergely, with " 10 x 10 patch of pixels ", I believe James means that he observes 100 neighbouring pixels each with 0 counts. Thus the frequentist view can be taken, and results in 0 as the variance, right?
best, Kay On Fri, 15 Oct 2021 21:07:26 +0000, Gergely Katona <gergely.kat...@gu.se> wrote: >Dear James, > >Uniform distribution sounds like “I have no idea”, but a uniform distribution >does not go from -inf to +inf. If I believe that every count from 0 to 65535 >has the same probability, then I also expect counts with an average of 32768 >on the image. It is not an objective belief in the end and probably not a very >good idea for an X-ray experiment if the number of observations are small. >Concerning which variance is the right one, the frequentist view requires >frequencies to be observed. In the absence of frequencies, there is no error >estimate. Bayesians at least can determine a single distribution as an answer >without observations and that will be their prior belief of the variance. >Again, I would avoid a uniform a priori distribution for the variance. For a >Poisson distribution the convenient conjugate prior is the gamma distribution. >It can control the magnitude of k and strength of belief with its location and >scale parameter, respectively. > >Best wishes, > >Gergely > >Gergely Katona, Professor, Chairman of the Chemistry Program Council >Department of Chemistry and Molecular Biology, University of Gothenburg >Box 462, 40530 Göteborg, Sweden >Tel: +46-31-786-3959 / M: +46-70-912-3309 / Fax: +46-31-786-3910 >Web: http://katonalab.eu, Email: gergely.kat...@gu.se > >From: CCP4 bulletin board <CCP4BB@JISCMAIL.AC.UK> On Behalf Of James Holton >Sent: 15 October, 2021 18:06 >To: CCP4BB@JISCMAIL.AC.UK >Subject: Re: [ccp4bb] am I doing this right? > >Well I'll be... > >Kay Diederichs pointed out to me off-list that the k+1 expectation and >variance from observing k photons is in "Bayesian Reasoning in Data Analysis: >A Critical Introduction" by Giulio D. Agostini. Granted, that is with a >uniform prior, which I take as the Bayesean equivalent of "I have no idea". > >So, if I'm looking to integrate a 10 x 10 patch of pixels on a weak detector >image, and I find that area has zero counts, what variance shall I put on that >observation? Is it: > >a) zero >b) 1.0 >c) 100 > >Wish I could say there are no wrong answers, but I think at least two of those >are incorrect, > >-James Holton >MAD Scientist >On 10/13/2021 2:34 PM, Filipe Maia wrote: >I forgot to add probably the most important. James is correct, the expected >value of u, the true mean, given a single observation k is indeed k+1 and k+1 >is also the mean square error of using k+1 as the estimator of the true mean. > >Cheers, >Filipe > >On Wed, 13 Oct 2021 at 23:17, Filipe Maia ><fil...@xray.bmc.uu.se<mailto:fil...@xray.bmc.uu.se>> wrote: >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) ><00006a19cead4548-dmarc-requ...@jiscmail.ac.uk<mailto:00006a19cead4548-dmarc-requ...@jiscmail.ac.uk>> > 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 ><and...@mrc-lmb.cam.ac.uk<mailto:and...@mrc-lmb.cam.ac.uk>> 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 ><ianj...@gmail.com<mailto:ianj...@gmail.com>> 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 ><jmhol...@lbl.gov<mailto:jmhol...@lbl.gov>> 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<http://www.jiscmail.ac.uk/CCP4BB>, a mailing list >hosted by www.jiscmail.ac.uk<http://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 > > >________________________________ > >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/ ######################################################################## 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/
