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I'm dealing with some biological data (read counts from gene expression measurements). For each individual we have 2 measurements, which correspond to two different version of the same gene. These measurements are discrete count data, and generally we would model them with with a negative binomial distribution, since they contain plenty of noise in a addition to the poisson distributed shot noise.

What I"m interested in is the ratio between counts. I'm given to understand that if we consider two poisson distributed random variables, and condition on their total counts, then the count for one in particular will be binomial distributed (correct me if I'm wrong). My question is this - what if we replace the poisson distributions here with negative binomial distributions? What is the distribution of the counts for each one, conditional on the total? Is it a beta binomial?

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    $\begingroup$ Do the two negative binomial distributions have the same parameter values? $\endgroup$ – jbowman Sep 26 '16 at 13:22
  • $\begingroup$ No, I don't have any reason to think they do, although they're likely to be quite similiar. $\endgroup$ – Dermot Harnett Sep 26 '16 at 15:01
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    $\begingroup$ Can these counts be 0? $\endgroup$ – Glen_b Sep 27 '16 at 4:43
  • $\begingroup$ The total must be at least 1, and will probably get thresholded higher than that at something like 20 or 50. $\endgroup$ – Dermot Harnett Sep 27 '16 at 15:30
  • $\begingroup$ ttu-ir.tdl.org/ttu-ir/bitstream/handle/2346/59954/… $\endgroup$ – kjetil b halvorsen♦ Oct 2 '16 at 19:07
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I don't have a formal proof, but I think your intuition is right and that it will be beta-binomial, under the assumption that the two Negative Binomials have the same "p" parameter. A hand-wavy explanation would be that Negative Binomial is just a Poisson with a gamma prior. If you have two Poissons and condition on the sum you get a Binomial. If you have two gammas with the same scale parameter and divide by the total you get a Beta.

More generally, if you have many Poissons and condition on the sum you get a Multinomial. If you have many Gammas with same scale and divide by the total you get a Dirichlet. So this suggests that if you take a bunch of Negative Binomials that have the same "p" parameter but possibly different "r" parameters and condition on their sum, it is likely to follow a Dirichlet-Multinomial distribution.

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