An economist wrote in, asking why it would make sense to fit Bayesian hierarchical models instead of frequentist random effects.
Short answer is that anything Bayesian can be done non-Bayesianly: just take some summary of the posterior distribution, call it an “estimator,” and there you go. Non-Bayesian can be regularized, it can use prior information, etc. No reason that a non-Bayesian method has to use p-values. To put it another way, there’s Ms. Bayesian and there’s Ms. Bayesian’s evil twin, who lives in a mirror world and does everything that Ms. Bayesian does, but says it’s non-Bayesian. The evil twin doesn’t trust Bayesian methods, she’s a real skeptic, so she just copies Ms. Bayesian but talks about regularizers instead of priors, and predictive distributions instead of posteriors. It doesn’t really matter, except that the evil twin might have more difficulty justifying her estimation choices because she can’t refer to a generative model.
Now if people want to defend some particular “frequentist” procedure, that’s another story. The procedures out there tend to under-regularize; they get noisy estimates of group-level variance parameters (see here and here) and they lead to overestimates of magnitudes of effect sizes (see here).
The usual non-Bayesian procedures are designed to work well asymptotically (in the case of hierarchical models, this is the limit as the number of groups approaches infinity). But as noted Bayesian J. M. Keynes could’ve said, asymptotically we’re all dead.