I’ve recently embarked on fitting multilevel regression models in the Bayesian framework, using a MCMC algorithm (brms in R actually).

I believe I have understood how to diagnose convergence of the estimation process (trace, geweke plot, autocorrelation, posterior distribution…).

One of the thing that strikes me in the Bayesian framework is that much effort seems to devoted to do those diagnostics, whereas very little appears to be done in terms of checking the residuals of the fitted model.

Long story short: I will probably have to present my model to the “classic” econometrician and he will expect me to discuss the residuals. Of course, there is a problem of defining “residuals” in Bayesian regression. So should I simply calculate the fitted model values (Estimate column from fitted(brm) function), replicate the multilevel model into LM and analyze the differences as typical residuals? Or should I focus on posterior predictive checks / loo validation?