Comments on: Poisson vs Gaussian, Part 2 http://hea-www.harvard.edu/AstroStat/slog/2009/poigauss-pdfs/ Weaving together Astronomy+Statistics+Computer Science+Engineering+Intrumentation, far beyond the growing borders Fri, 01 Jun 2012 18:47:52 +0000 hourly 1 http://wordpress.org/?v=3.4 By: vlk http://hea-www.harvard.edu/AstroStat/slog/2009/poigauss-pdfs/comment-page-1/#comment-879 vlk Wed, 15 Apr 2009 21:31:20 +0000 http://hea-www.harvard.edu/AstroStat/slog/?p=2181#comment-879 I'm afraid you are reading too much into it. Given data in the form of counts, you can calculate the exact posterior pdf for the source intensity based on the Poisson likelihood. That gives one curve. You can also make an approximation and compute the equivalent Gaussian MLE and variance. That gives another curve. Compare, contrast, conclude. I’m afraid you are reading too much into it. Given data in the form of counts, you can calculate the exact posterior pdf for the source intensity based on the Poisson likelihood. That gives one curve. You can also make an approximation and compute the equivalent Gaussian MLE and variance. That gives another curve. Compare, contrast, conclude.

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By: hlee http://hea-www.harvard.edu/AstroStat/slog/2009/poigauss-pdfs/comment-page-1/#comment-878 hlee Wed, 15 Apr 2009 19:33:15 +0000 http://hea-www.harvard.edu/AstroStat/slog/?p=2181#comment-878 As I kept confused with astronomer's "model uncertainty" although I know it is used differently from statistician's "model uncertainty", perhaps pdf is one of such. Poisson pdf and Gaussian pdf have their own equation formats, so in probability Poisson pdf cannot be written in terms of Gaussian pdf; however, in distribution, asymptotically via joint distributions or likelihoods (not a function of single datum X, but a function of X_1,...,X_n), as if done with the central limit theorem, the empirical distribution of Poisson data can be represented by Normal distribution. I read the first sentence as "A pdf (a equation of a certain class, defined on the space of positive integers) is summarized by the other equation (Gaussian pdf, symmetric, spans from -infty to infty)" when a Poisson pdf is not comparable to a normal pdf. The reason of introducing the Berry-Esseen is that it proves such gap is inevitable in approximation (doesn't matter whether a sampling distribution is Poisson, Gamma, binomial, etc) and it tells that there's lower bound in gaps although case by case the gap sizes would be different let alone sample sizes. One would surprise how large n can be in order to make the gap small enough to a tolerable level. Note that it assumes iid and poses some constraints on moments. Theorems are there to be modified by relaxing conditions and assumptions, say from iid to independent, to prove other problems. Such tailoring yields a new theorem with another name or corollaries. Instead of saying >25 is a golden absolute rule and 10> is wrong in X-ray spectral grouping (counts in bins), I'd like to say more care is needed when data are binned/grouped. It's hard to know those counts in each group satisfy these assumptions and produce small enough gap of tolerance for traditional the x-sigma error bar construction based on the delta chi method. Fortunately, there are many efforts to minimize the impact of gaps in your plots with low count data while lessening sacrifice of innate information. They look so far just rules of thumb without mathematical justifications; at least documentations from XSPEC and Sherpa gave me such impression (some literatures take 20 intead of 25, suggest averaging errors of neighbors but do not provide definition/function/explanation to define degrees of neighborhood, etc). Quite room for mathematical statisticians to accompany astronomers' lonely quest and to guide as well. As I kept confused with astronomer’s “model uncertainty” although I know it is used differently from statistician’s “model uncertainty”, perhaps pdf is one of such. Poisson pdf and Gaussian pdf have their own equation formats, so in probability Poisson pdf cannot be written in terms of Gaussian pdf; however, in distribution, asymptotically via joint distributions or likelihoods (not a function of single datum X, but a function of X_1,…,X_n), as if done with the central limit theorem, the empirical distribution of Poisson data can be represented by Normal distribution. I read the first sentence as “A pdf (a equation of a certain class, defined on the space of positive integers) is summarized by the other equation (Gaussian pdf, symmetric, spans from -infty to infty)” when a Poisson pdf is not comparable to a normal pdf.

The reason of introducing the Berry-Esseen is that it proves such gap is inevitable in approximation (doesn’t matter whether a sampling distribution is Poisson, Gamma, binomial, etc) and it tells that there’s lower bound in gaps although case by case the gap sizes would be different let alone sample sizes. One would surprise how large n can be in order to make the gap small enough to a tolerable level. Note that it assumes iid and poses some constraints on moments. Theorems are there to be modified by relaxing conditions and assumptions, say from iid to independent, to prove other problems. Such tailoring yields a new theorem with another name or corollaries. Instead of saying >25 is a golden absolute rule and 10> is wrong in X-ray spectral grouping (counts in bins), I’d like to say more care is needed when data are binned/grouped. It’s hard to know those counts in each group satisfy these assumptions and produce small enough gap of tolerance for traditional the x-sigma error bar construction based on the delta chi method.

Fortunately, there are many efforts to minimize the impact of gaps in your plots with low count data while lessening sacrifice of innate information. They look so far just rules of thumb without mathematical justifications; at least documentations from XSPEC and Sherpa gave me such impression (some literatures take 20 intead of 25, suggest averaging errors of neighbors but do not provide definition/function/explanation to define degrees of neighborhood, etc). Quite room for mathematical statisticians to accompany astronomers’ lonely quest and to guide as well.

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By: vlk http://hea-www.harvard.edu/AstroStat/slog/2009/poigauss-pdfs/comment-page-1/#comment-875 vlk Wed, 15 Apr 2009 10:33:42 +0000 http://hea-www.harvard.edu/AstroStat/slog/?p=2181#comment-875 It's just the <a href="http://hea-www.harvard.edu/AstroStat/slog/2008/eotw-bkgsubtract-poisson/" rel="nofollow">pdf</a>, the joint posterior of source and background marginalized over background. What's contradictory about it? It’s just the pdf, the joint posterior of source and background marginalized over background. What’s contradictory about it?

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By: hlee http://hea-www.harvard.edu/AstroStat/slog/2009/poigauss-pdfs/comment-page-1/#comment-874 hlee Wed, 15 Apr 2009 05:33:01 +0000 http://hea-www.harvard.edu/AstroStat/slog/?p=2181#comment-874 The first sentence, because of "probability density function," sounds self contradictory to me. If it's replaced by "empirical distribution function" it seems o.k. but I'm a bit cautious to assure it since I couldn't find literature in astronomy that used "empirical cdf." I'll get back to the issue why I felt uncomfortable with the first sentence later in my [MADS]. I just wanted to let astronomers know <a href="http://en.wikipedia.org/wiki/Berry–Esséen_theorem" rel="nofollow">Berry–Esséen theorem (wiki link).</a> A brief self defense for the uncomfortable feeling is that Poisson and Gaussian both has its own probability density function, which belongs to an exponential family and I guess F_n(x) in the Berry-Esseen is related to your "probability density function." The first sentence, because of “probability density function,” sounds self contradictory to me. If it’s replaced by “empirical distribution function” it seems o.k. but I’m a bit cautious to assure it since I couldn’t find literature in astronomy that used “empirical cdf.” I’ll get back to the issue why I felt uncomfortable with the first sentence later in my [MADS]. I just wanted to let astronomers know Berry–Esséen theorem (wiki link). A brief self defense for the uncomfortable feeling is that Poisson and Gaussian both has its own probability density function, which belongs to an exponential family and I guess F_n(x) in the Berry-Esseen is related to your “probability density function.”

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