If we do a negative binomial GLM with unknown deviance, a frequent strategy (used for example by
glm.nb in package
MASS in R) is to use a Gibbs sampler:
- Hold dispersion fixed, estimate the mean
- Hold mean fixed, estimate the dispersion
Isn't the mean the same than that would be guessed by Poisson regression? If so, why bother with the Gibbs sampler? Why not just fit the mean using Poisson, then fit the dispersion?