I’ve got to figure out what people mean when they say “fixed effect” and “random effect”, because I’ve been confused about it for a year and I’ve been hearing it all the time lately.

Bayesian Data Analysis is my starting guide, which includes a footnote on page 391:

The terms ‘fixed’ and ‘random’ come from the non-Bayesian statistical tradition are are somewhat confusing in a Bayesian context where all unknown parameters are treated as ‘random’. The non-Bayesian view considers fixed effects to be fixed unknown quantities, but the standard procedures proposed to estimate these parameters, based on specified repeated-sampling properties, happen to be equivalent to the Bayesian posterior inferences under a noninformative (uniform) prior distribution.

That doesn’t totally resolve my confusion, though, because my doctor-economist colleagues are often asking for the posterior mean of the random effects, or similarly non-non-Bayesian sounding quantities.

I was about to formulate my working definition, and see how long I can stick to it, but then I was volunteered to teach a seminar on this very topic! So instead of doing the work now, I turn to you, wise internet, to tell me how I can understand this thing.