Matt Wand asks,
I’m wondering if you have any pointers to putting priors on shape parameters (e.g. the positive one that extends the Poisson to negative binomial)? I’ve already started taking advice for variance components from you.
My thoughts:
As implied by Matt’s question, shape parameters are closely connected to hierarchical variance parameters. For example, the negative binomial shape parameter can be mapped to the coefficient of variation of a latent gamma distribution for the underlying probabilities. Thus, one approach would be to put a prior distribution on this coefficient of variation parameter–which is very close to a hierarchical sd–and then transform that back to an implication for the negative binomial shape parameter.
In our paper on social networks, we used uniform prior distributions on the negative binomial overdispersion parameters, and I think we could’ve done better, in this case using a hierarchical model on these parameters. (We had 32 of them in our example.) This is the approach recommended in Section 6 of our paper.
P.S. See Dana’s suggestion in comments: it’s a model that seems to allow the distribution to be underdispersed, which is slightly different (maybe better) than my model. Dana suggests an exponential prior distribution for the coefficient of variation parmater, which seems like it could work, but I have a sentimental attachment to the half-Cauchy because it is flat near zero and allows for occasional large values.
We've had pretty good success reparameterizing the gamma-Poisson (negative binomial) distribution in terms of the mean and coefficient of variation (ratio of standard deviation to mean). In this case, the CV equals the reciprocal of the square root of the shape parameter. We expect the CV to be near unity, but with significant uncertainty, so we chose an exp(1) prior for the CV. This can be justified formally as a maximum entropy prior for the CV.