Neal writes,

In your entry on mutlilevel modeling, you note that de Leeuw was “pretty critical of Bayesian multilevel modeling” In your paper, you say “compared with classical regression, multilevel model is almost an improvement, but to varying degrees.”

So my question to you is: other than issues of computations, and perhaps not jumping linguistic hoops, what is the relevance of the Bayesian modifier of multilevel modeling? Would the issues be any different for classical mixed effects modeling?

My response: the Bayesian version averages over uncertainty in the variance parameters. This is particularly important when the number of groups is small, or the model is complicated, and when the actual group-level variance is small, in which case it can get lost in the noise.

Also, we discuss some of this in Sections 11.5 in our book. I hope we made the above point somewhere in the book, but I’m not sure that we remembered to put it in. The point is made most clearly (to me) in the 8-schools example, which is in Chapter 5 of Bayesian Data Analysis and comes from an article by Don Rubin from 1981.