Thanks for the suggestion, @Dekermanjian! I did consider that, but I’m not sure it makes sense in my case. Since I’m working with count data, scaling it directly could mess with one of its key properties—namely, that it’s discrete.
That said, because Negative Binomial regression uses a log link, the data is effectively scaled in the model. In my case, log(mu) falls roughly between 3 and 7, and I’ve tried to reflect that in how I set up my priors.