# Marginal Dirichlet Negative Binomial Distribution and the Multinomial Inverse Polya Urn

I have the following 'URN-like' problem - assume an urn the contains balls with m different colors. As in the standard Polya scheme, every time a ball is sampled, it is returned the urn in addition with another ball of the same color. This is the standard multivariate Polya urn process. Let's assume that one of the colors (e.g., red) is defined as a 'success'. I am interested in finding the PMF (and eventually also the CMF) of the number of total failures (non red balls) before k red balls are sampled.

The Dirichlet negative multinomial distribution defines the multivariate PMF over the entire vector of non-red colors. I think that what I need is called the 'marginal distribution'. Wikipedia says that: "To obtain the marginal distribution over a subset of Dirichlet negative multinomial random variables, one only needs to drop the the variables that one wants to marginalize out from the $$\alpha$$ vector..." - but I cannot understand the meaning of that (particularly for the case in which I want to marginalize out all 'failure variables').

I know that in the case of the Dirichlet multinomial distribution (addressing the distribution of the number of successes given a known n) marginalizing out all failures simply reduces to the beta-binomial distribution with parameters $$(\alpha_{success}, \sum{\alpha_{failures},n})$$. So can this be used to infer that I can also use the Dirichlet negative binomial distribution with parameters $$(\alpha_{success}, \sum{\alpha_{failures},k})$$?

Thanks a lot!

Isaac

A possible source of confusion is that the Dirichlet negative multinomial provides the joint distribution of all $$m$$ colors - not including the stopping color. In your case, it sounds like you are defining the stopping color as red and are stopping when $$k$$ red balls are drawn. However, popular choices for notation in the number for the stopping color are "$$r$$" (Wikipedia article on negative binomial) and "$$x_0$$" (Wikipedia article on Dirichlet Negative Multinomial).
It also does not sound like you are interested in the joint distribution of all non-red balls. Rather, it sounds as if you are interested in the total number of non-red ball draws and that would mean that you want the univariate version of the Dirichlet negative multinomial - namely the beta-negative binomial. The Wikipedia article on the beta-negative binomial uses parameters ($$r$$, $$\alpha$$, $$\beta$$) .
Also note the difference between "marginalization" and "aggregation". You are appealing to the aggregation property of Dirichlet multinomial in your last paragraph. The aggregation property does indeed hold for the Dirichlet negative multinomial as well. Perhaps another source of confusion is that it is also true that the marginalization involves a beta-binomial distribution. In particular, it is also true that the marginal distribution for the number of failures of Type 1 is beta-binomial with parameters $$(\alpha_{success}, \alpha_{Type 1 failure}, n)$$. And this type of marginalization property does indeed hold for the Dirichlet negative multionmial as well.
The distribution of non-red balls draws in a literal Pólya urn model of the type you discribed (where 1 additional ball of the same color is returned to the urn) is beta-negative binomial with parameters ($$r$$, $$\alpha$$, $$\beta$$) where $$r$$ is the number of red balls observed to be drawn and used to determine stopping (you called it $$k$$), $$\alpha$$ and $$\beta$$ are the initial number of red balls and non-red balls respectively in the urn.