This is a nice clarification, but then shouldn’t “a posterior distribution of the expectation of the (draw-wise conditional) posterior predictive distribution." instead reference “expectation s” (plural)? The confusing part about the brms documentation is then that it is it does seem to imply there is a single posterior predictive for each draw.
@scholz Just looked and Rethinking explains what the posterior predictive is more clearly (but does not discuss the expected value of the distribution vs. the distribution of the expected values issue.
@avehtari The discussion in Regression and Other Stories describes epred (when passed new data) as “the expected prediction for a new data point” (p.116) and says "we can compute uncertainty in the expected value or predicted probability E(y_{new}) = logit^{-1}(x_{new} \beta) " (p.223) which seems to imply the posterior of the expected value. Similarly, the same type of description is implied on p.315.