I am fitting a hierarchical GLM to replicates of the same experiment as bayesian version of a mixed effects linear model, such that there are hyperpriors for the “regular” priors, which in turn shared by replicate-specific coefficients. It boils down to something like this:

Now \beta_1 and \beta_2 may even have opposite signs, and I want to make a statement about the effect of the \beta coefficients in general, not the replicate-specific \beta_i, and since they are related by the shared prior I am thinking that I can refer to that distribution instead.

Is there a standard interpretation for what the estimates for \mu and \sigma (and their posteriors) mean, and how to interpret that in relationship to the actual parameters to which they define a prior? Beyond the vast literature on hierarchical models is there some specific literature on the interpretation of those priors?

Yes. Here \mu is the group-level mean and \sigma is the group-level standard deviation. What “group” means will depend on the specifics of your problem. I don’t understand what you mean by “actual” parameters which define the prior. These are hyperparameters, that index the probability distribution of random quantities in the model.

Sorry about that sentence, it should be “to the actual parameters to which they define a prior”, not “the define…”. I edited the post because that sentence didn’t make sense as it was.

I understand that the \mu and \sigma are the parameters that define the group-level distribution, and I guess you are right that it will always depend on the specific problem. Here, the problem is that the replicates are a sort of nuisance parameter we’d like to “integrate over” in a sense. I think for that purpose, the statement I want to make is really about the group-level parameters and the replicate-specific ones can be essentially ignored, but I see how there may be other problems where the individual-level parameters may be just as important. Thanks.