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

]]>http://newton.hep.upenn.edu/~heinrich/birs/challenge.pdf

Is that of interest?

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[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]

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.

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[Response:I've already saidsis the parameter of interest. There is nothing that saysscannot be 0. There is an integrable posterior probability distributionp(s|N_S,N_B)onsover [0,+\infty]. N_S aredata. 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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[Response:Boundary of what?

To clarify,sis 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 forsand the background intensityb; andp(s|N_S,N_B)is the background marginalized posterior probability distribution fors. See Section 2.1 of van Dyk et al. (2001, ApJ 548, 224) --sis the same as lambda^S, etc. in their Eqn 5, 6.

-vlk]

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[Response:No, the posterior is not improper (as long asa>0). Nor is it misspecified in any sense that I know of. You cannot mix posterior distributions obtained fromdifferentdata. 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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[Response:There is no confusion with the value of N_S -- these are theobserveddata. So I can't say where that mixture model will apply.

-vlk]