Posterior prediction terminology (hierarchical model)

General
1

Hi all,

I am wondering about the right terminology for two kinds of posterior predictions/estimations I am making with a hierarchical model (and whether these predictions make sense in the first place).

In my model I have population and subject-level parameters. I have a parameter q to capture the “population baseline” (q_mu), a parameter to capture deviations according to subgroups in my data (q_group) and a parameter for each subject i (q_i), where q_i = q_mu + q_group[i] + subject-level effect.

I have two approaches for estimating posterior medians for different subgroups in my data:

  1. I group subjects according to subgroup and then compute medians from subject-level parameters (q_i) in the respective groups (first computing the median by posterior draw, then the overall median).

  2. I compute the posterior medians for each group by completely ignoring the subject-level effect (so just q_mu + q_group[j] with j indexing the group I’m interested in), computing the median over posterior draws.

I’m wondering if a) the second approach is even valid, and b) what would you call these kinds of estimations? Something like “Posterior medians computed from subject-level parametes” and “Population-level posterior medians”, respectively? I’m at a bit of a loss here. Also, would you even call these “posterior predictions”? I guess they’re not really predictions (no new input data).

Thank you!