By the way, I find this text very useful and illuminating, in many ways. I came to it through its extension of the Fisher Exact Test to instances where the extended hypergeometric distribution is more appropriate for tables of counts, where the model suggests the populations compared may not have the same probabilities of appearance. There’s a lot more to this book.

I have also looked at both Fienberg, The Analysis of Cross-Classified Categorical Data (2nd edition, Springer, 2007) , which is useful, and Congdon’s Bayesian Models for Categorical Data, Wiley, 2005. I find Congdon less useful, even if he incorporates some of Epstein and Fienberg, “Bayesian estimation in multidimensional contingency tables” (from Computer Science and Statistics: Proceedings of the 23rd Symposium on the Interface, Keramidas, E, editor, Inteface Foundation of North America: Fairfax, 37-47).

Hope this helps.

There are a wide variety of NB models, including one that is not a Poisson-gamma mixture, but rather derives directly from PDF as understood by the first version mentioned by John. I have called this the NB-C parameterization, and it is not related to the Poisson at all. On the other hand, with the variance defined as mu+alpha*mu^2, where alpha is the overdispersion parameter and mu is the mean, the traditonal NB, called in the literature as NB2, is parameterized such that the Poisson model is a NB with alpha=0. A geometric model is NB with alpha=1. A NB2 model can be extra-dispersed as well — either uncer of over dispersed. The data can in fact be Poisson overdispersed but NB under-dispersed. How to identify possible overdispersion (correlation) and its causes is important in deciding which type of count model to use for a given data situation. If you are really interested in this subject, email me and I can perhaps refer you to sources that will help you understand the family of negative binomial models a bit better. Hilbe@asu.edu

VAR[...] = p(1) v(1) + p(2) v(2) + p(1) p(2) [u(1) - u(2)]^2.

The question is, what does this look like in general? It had been suggested to me that the general expression, given the intermediate definition,

Y(i) = P(S == i) X(i)

for the variance looks like

SUM-over-i VAR[Y(i)] + 2 SUM-over-i-not-equal-j COV[Y(i),Y(j)]

where, of course, COV[U,V] = E[U V] – E[U] E[V]

Here, since Y(i) and Y(j) are mutually exclusive, E[Y(i) Y(j)] is presumably zero.

But I cannot reconcile this in the case of two r.v. with n = 2 expression.

Any suggestions?