# What is a “batch” of coefficients in Bayesian multilevel modeling?

I’m well acquainted with frequentist approaches for multilevel models (i.e. mixed/random effects models with random intercepts or slopes), and empirical Bayes estimation, but I’m trying to get familiar with true Bayesian multilevel modeling using Andrew Gelman’s book and papers, but there a phrase he keeps repeating that I don’t understand and that no one else seems to use.

In this paper Gelman et al lay out a multilevel logit model to estimate support for some candidate in different states. The respondent level model includes the following variables: female, race (black vs anything else), an interaction between female and black, age (four categories), and education (four categories). After explaining this they then say:

“As part of a general approach for multilevel modeling, we give each batch of regression coefficients with greater than two groups an independent normal distribution centered at 0 with standard deviation estimated from data. This allows us to estimate these parameters as varying effects, taking advantage of the multilevel structure of the data. There is no gain to multilevel modeling for batches with J<3 groups when prior distributions are uninformative so for simplicity we model sex and ethnicity as regression coefficients with no multilevel structure.”

My first question is: What do they mean by a “batch” of regression coefficients? Based on the last sentence my guess is that they’re talking about what I would call dummy variables: If I wanted to control for a 4 category education variable in my model I’d include 3 binary "dummy" variables for (say) high school grad, college grad, and graduate degree, with “didn’t finish high school” as the omitted category. Is that what they’re talking about here?

Secondly: If that’s so then what does grouping individual level coefficients into “batches” have to do with multilevel modeling specifically? As I understand it the “multilevel” aspect of this analysis concerns the nesting of individuals within states and the specification of a separate state-level model, all of which makes sense to me. But the entire discussion of “batches” seems to only concern data at the individual level, and has nothing to do with state. So why is is assigning normal distributions to “batches” of individual level coefficients like race and gender “multilevel modeling”?

Thirdly: Why is there no benefit to multilevel modeling when a “batch” has fewer than 3 groups? (Maybe this will be obvious once I know the answer to my other questions)

I see that Andrew Gelman uses the phrase “batches of coefficients” in lots of different discussions of multilevel modeling, but none of these discussions have made it clear to me what exactly he’s talking about and no one else seems to use this terminology as far as I can tell.