How many parameters are in a multilevel model?

Stephen Collins writes:

I’m reading your Multilevel modeling book and am trying to apply it to my work. I’m concerned with how to estimate a random intercept model if there are hundreds/thousands of levels. In the Gibbs sampling, am I sampling a parameter for each level? Or, just the hyper-parameters? In other words, say I had 500 zipcode intercepts modeled as ~ N(m,s). Would my posterior be two dimensional, sampling for “m” and “s,” or would it have 502 dimensions?

My reply: Indeed you will have hundreds or thousands of parameters—or, in classical terms, hundreds or thousands of predictive quantities. But that’s ok. Even if none of those predictions is precise, you’re learning about the model.

See page 526 of the book for more discussion of the number of parameters in a multilevel model.