In this case study, the LOO-adjusted R^2 uses the Bayesian bootstrap (in addition to the LOO-CV-based predictions \hat{y}_{\text{loo},n}). In contrast, this answer and brms seem to omit the Bayesian bootstrap.

The case study from above suggests (but not explicitly says) that the Bayesian bootstrap accounts for the fact that the true data-generating distribution (for y) is unknown. But couldn’t one argue that the true data-generating distribution is reflected by the observed data? If yes, then wouldn’t the Bayesian bootstrap introduce additional *sampling* uncertainty (*sampling* in the frequentist sense of repeating the data observation process)? If this is correct, then wouldn’t it make sense to omit the Bayesian bootstrap (like in the answer linked above and in brms)?

So my question is: Is it *incorrect* to use the Bayesian bootstrap for the LOO-adjusted R^2 or is the Bayesian bootstrap *optional* for the LOO-adjusted R^2?