Default prior on the negative binomial is very informative

Summary:

The default prior for the over-dispersion parameter of the negative binomial likelihood puts a lot of prior mass on large amounts of over-dispersion

Description:

This is found in src/stan_files/count.stan. The likelihood is invoked on lines 99-100 and the prior on the variable aux is set on lines 108-117.

The variance of the neg_binomial2 is given by
variance = mean(1 + mean / aux)
where aux has a half-normal, half-t, or exponential prior. This means that most of the prior mass is on aux<1 which can lead to a great deal of over-dispersion.

The fix is to put the same prior on 1/aux or, even better, 1/sqrt(aux). The inverse square root comes from noting that you can specify a negative binomial as a poisson with a random mean with a Gamma(aux,aux) distribution. This has mean 1 and variance 1/aux. This suggests that 1/sqrt(aux) is somewhat like a standard deviation.

An outline of a different, more mathematical approach based on PC priors and Kullback-Leibler divergences is given in the second last section (Can't help lovin that man) of this blog post: http://andrewgelman.com/2018/04/03/justify-my-love/

Reproducible Steps:

Not applicable

RStanARM Version:

Not applicable. This is true on the current master branch https://github.com/stan-dev/rstanarm/commit/f11d6bf92b6c78ce64896caf78c2237ebd015abb

R Version:

Not applicable

Operating System:

Not applicable

Start in src/stan_files/count.stan, especially the negative-binomial likelihood on lines 99-100 and the aux prior on lines 108-117. Review the discussion and the proposed parameterizations before choosing the intended default; done means the prior is changed consistently and the resulting over-dispersion behavior is validated.