Comments on: When you observed zero counts, you didn’t not observe any counts http://hea-www.harvard.edu/AstroStat/slog/2007/zero-counts/ 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: hlee http://hea-www.harvard.edu/AstroStat/slog/2007/zero-counts/comment-page-1/#comment-106 hlee Wed, 03 Oct 2007 19:37:02 +0000 http://hea-www.harvard.edu/AstroStat/slog/2007/zero-counts/#comment-106 <p>I'm ready to take any blames and admit my ignorance (I only can propose approaches limited to my knowledge and readings from the slog). I wonder if there's a Bayesian counter part of Quantile Regression. Such model could give the upper limit at N_s=0 by getting the corresponding quantiles.</p> <p>[After talking to Vinay] I misunderstood the objectivity of Vinay's question. But, I hope Poisson Quantiles may assist low count data analysis.</p> I’m ready to take any blames and admit my ignorance (I only can propose approaches limited to my knowledge and readings from the slog). I wonder if there’s a Bayesian counter part of Quantile Regression. Such model could give the upper limit at N_s=0 by getting the corresponding quantiles.

[After talking to Vinay] I misunderstood the objectivity of Vinay’s question. But, I hope Poisson Quantiles may assist low count data analysis.

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By: vlk http://hea-www.harvard.edu/AstroStat/slog/2007/zero-counts/comment-page-1/#comment-101 vlk Sun, 30 Sep 2007 02:30:41 +0000 http://hea-www.harvard.edu/AstroStat/slog/2007/zero-counts/#comment-101 Paul, yes, the behavior described in section 8.3 in that BIRS writeup is exactly what I wrote about above. There are two items in the Heinrich draft that I do not quite understand though -- first, why the parenthetical insistence on 0 background events? and second, what does he mean by "absolute separation" between source and background? Paul, yes, the behavior described in section 8.3 in that BIRS writeup is exactly what I wrote about above. There are two items in the Heinrich draft that I do not quite understand though — first, why the parenthetical insistence on 0 background events? and second, what does he mean by “absolute separation” between source and background?

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By: Paul B http://hea-www.harvard.edu/AstroStat/slog/2007/zero-counts/comment-page-1/#comment-100 Paul B Sun, 30 Sep 2007 00:29:45 +0000 http://hea-www.harvard.edu/AstroStat/slog/2007/zero-counts/#comment-100 What is the actual model you are both talking about? Without knowing the details, it sounds like a similar situation to the Banff model, discussed on pg15 of: http://newton.hep.upenn.edu/~heinrich/birs/challenge.pdf Is that of interest? <blockquote><b>[Response:</b> It is a subset of the Banff problem, one that does not include the 3rd equation. i.e., N_S ~ Pois(<em>s</em>+<em>b</em>) N_B ~ Pois(<em>t*b</em>) See Eqns 5 and 6 of van Dyk et al. 2001 (linked in above). It is the standard "background subtraction" problem in [X-ray] astronomy, where you need to infer the intensity of a source from two measurements: one of background only, and one of source+background. In the high counts limit, E(<em>s</em>) = N_S - (N_b/t) . -vlk<b>]</b></blockquote> What is the actual model you are both talking about? Without knowing the details, it sounds like a similar situation to the Banff model, discussed on pg15 of:
http://newton.hep.upenn.edu/~heinrich/birs/challenge.pdf
Is that of interest?

[Response: It is a subset of the Banff problem, one that does not include the 3rd equation. i.e.,
N_S ~ Pois(s+b)
N_B ~ Pois(t*b)
See Eqns 5 and 6 of van Dyk et al. 2001 (linked in above).

It is the standard "background subtraction" problem in [X-ray] astronomy, where you need to infer the intensity of a source from two measurements: one of background only, and one of source+background. In the high counts limit,
E(s) = N_S – (N_b/t) .
-vlk]

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By: hlee http://hea-www.harvard.edu/AstroStat/slog/2007/zero-counts/comment-page-1/#comment-99 hlee Thu, 27 Sep 2007 23:05:56 +0000 http://hea-www.harvard.edu/AstroStat/slog/2007/zero-counts/#comment-99 If θ is the parameter of your interest, {θ:θ>0} is an open set, which specifies the parameter space. Openness and closeness also specifies the data space. When I read your posting, my impression was at N_s=0 (the boundary), the posterior behaves in an unexpected way, which led me to suggest a mixture model to resolve the strange behavior. My specifying distribution properly implies that the distribution behave properly in the data/parameter space, including N_s=0. If the boundary N_s=0 causing troubles, then keep the current posterior for N_s>0 and add another component from different family for N_s=0 (the boundary). [Checking the validity of this mixture model is another topic.] I'd rather point a well cited paper, to indicate what I meant by misspecified. <a href="http://www.jstor.org.ezp1.harvard.edu/view/00129682/di952657/95p0102c/0" rel="nofollow">Maximum Likelihood Estimation of Misspecified Models</a> by H. White (1982) in Econometrica. References therein are quite classical. Mixture models and mixing are different topics, I guess. I only know a bit of mixture models. I hope mixing didn't occur to you. <blockquote><b>[Response:</b> I've already said <em>s</em> is the parameter of interest. There is nothing that says <em>s</em> cannot be 0. There is an integrable posterior probability distribution <em>p(s|N_S,N_B)</em> on <em>s</em> over [0,+\infty]. N_S are <strong>data</strong>. When N_S=0, that is your measurement, which is one number, and is fixed, unchanging, immutable, for that observation. I suspect that you may be confusing a Bayesian calculation with frequentist ideas. -vlk<b>]</b></blockquote> If θ is the parameter of your interest, {θ:θ>0} is an open set, which specifies the parameter space. Openness and closeness also specifies the data space. When I read your posting, my impression was at N_s=0 (the boundary), the posterior behaves in an unexpected way, which led me to suggest a mixture model to resolve the strange behavior. My specifying distribution properly implies that the distribution behave properly in the data/parameter space, including N_s=0. If the boundary N_s=0 causing troubles, then keep the current posterior for N_s>0 and add another component from different family for N_s=0 (the boundary). [Checking the validity of this mixture model is another topic.]
I’d rather point a well cited paper, to indicate what I meant by misspecified.
Maximum Likelihood Estimation of Misspecified Models by H. White (1982) in Econometrica. References therein are quite classical.
Mixture models and mixing are different topics, I guess. I only know a bit of mixture models. I hope mixing didn’t occur to you.

[Response: I've already said s is the parameter of interest. There is nothing that says s cannot be 0. There is an integrable posterior probability distribution p(s|N_S,N_B) on s over [0,+\infty]. N_S are data. When N_S=0, that is your measurement, which is one number, and is fixed, unchanging, immutable, for that observation. I suspect that you may be confusing a Bayesian calculation with frequentist ideas.
-vlk]

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By: hlee http://hea-www.harvard.edu/AstroStat/slog/2007/zero-counts/comment-page-1/#comment-98 hlee Thu, 27 Sep 2007 15:54:37 +0000 http://hea-www.harvard.edu/AstroStat/slog/2007/zero-counts/#comment-98 Well, then your posterior is built on an open set. No reason for considering the boundary. <blockquote><b>[Response:</b> Boundary of what? To clarify, <em>s</em> is the source intensity, and is the parameter of interest; N_S and N_B are the observed data; <em>p(sb|N_S,N_B)</em> is the joint posterior for <em>s</em> and the background intensity <em>b</em>; and <em>p(s|N_S,N_B)</em> is the background marginalized posterior probability distribution for <em>s</em>. See Section 2.1 of <a href="http://adsabs.harvard.edu/abs/2001ApJ...548..224V" rel="nofollow">van Dyk et al. (2001, ApJ 548, 224)</a> -- <em>s</em> is the same as lambda^S, etc. in their Eqn 5, 6. -vlk<b>]</b></blockquote> Well, then your posterior is built on an open set. No reason for considering the boundary.

[Response: Boundary of what?
To clarify, s is the source intensity, and is the parameter of interest; N_S and N_B are the observed data; p(sb|N_S,N_B) is the joint posterior for s and the background intensity b; and p(s|N_S,N_B) is the background marginalized posterior probability distribution for s. See Section 2.1 of van Dyk et al. (2001, ApJ 548, 224) -- s is the same as lambda^S, etc. in their Eqn 5, 6.
-vlk]

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By: hlee http://hea-www.harvard.edu/AstroStat/slog/2007/zero-counts/comment-page-1/#comment-97 hlee Wed, 26 Sep 2007 22:12:10 +0000 http://hea-www.harvard.edu/AstroStat/slog/2007/zero-counts/#comment-97 Does the response imply the posterior distribution is improper? Does it mean the posterior distribution is misspecified? Then, instead of mixture, looking for other distributions for a confidence bound. The point of mixture is to keep the current posterior when N_s>0 but to add an additional component for N_s=0 to avoid the behavior you've described. r could be an indicator, I(N_s=0). I admit that I didn't know N_s is deterministic. I thought you only observe N, hardly separable in two, N_s and N_b, in a deterministic way. <blockquote><b>[Response:</b> No, the posterior is not improper (as long as <em>a</em>>0). Nor is it misspecified in any sense that I know of. You cannot mix posterior distributions obtained from <em>different</em> data. N_S are the counts in the source region, N_B in the "off-source", aka background, region. When you observe 0 counts in the source region, that is what you observe, there is no other N_S to mix it with. -vlk<b>]</b></blockquote> Does the response imply the posterior distribution is improper? Does it mean the posterior distribution is misspecified? Then, instead of mixture, looking for other distributions for a confidence bound. The point of mixture is to keep the current posterior when N_s>0 but to add an additional component for N_s=0 to avoid the behavior you’ve described. r could be an indicator, I(N_s=0). I admit that I didn’t know N_s is deterministic. I thought you only observe N, hardly separable in two, N_s and N_b, in a deterministic way.

[Response: No, the posterior is not improper (as long as a>0). Nor is it misspecified in any sense that I know of. You cannot mix posterior distributions obtained from different data. N_S are the counts in the source region, N_B in the "off-source", aka background, region. When you observe 0 counts in the source region, that is what you observe, there is no other N_S to mix it with.
-vlk]

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By: hlee http://hea-www.harvard.edu/AstroStat/slog/2007/zero-counts/comment-page-1/#comment-96 hlee Wed, 26 Sep 2007 15:01:11 +0000 http://hea-www.harvard.edu/AstroStat/slog/2007/zero-counts/#comment-96 Without knowing the physics of choosing gamma distribution, my question may look nonsensical but I wonder if it's possible to build a mixture model, r*p(s|N_s=0,N_b)+(1-r)*p(s|N_s>0,N_b). The first component comes from other type of distribution, not gamma. I believe astronomers know how to infer r. If it's a single case and not sure N_s=0 or N_s>0, you can select between two (r=1 or 0) based on model selection criteria or bayes factor. Then you could find proper upper limits based on post model selection inference. <blockquote><b>[Response:</b> There is no confusion with the value of N_S -- these are the <em>observed</em> data. So I can't say where that mixture model will apply. -vlk<b>]</b></blockquote> Without knowing the physics of choosing gamma distribution, my question may look nonsensical but I wonder if it’s possible to build a mixture model, r*p(s|N_s=0,N_b)+(1-r)*p(s|N_s>0,N_b). The first component comes from other type of distribution, not gamma. I believe astronomers know how to infer r. If it’s a single case and not sure N_s=0 or N_s>0, you can select between two (r=1 or 0) based on model selection criteria or bayes factor. Then you could find proper upper limits based on post model selection inference.

[Response: There is no confusion with the value of N_S -- these are the observed data. So I can't say where that mixture model will apply.
-vlk]

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