I'm trying to bootstrap non-parametric prediction errors for a model I'm building. I've seen a few resources that suggest that the proper procedure is the following, with the input matrix $X$, response vector $y$, an input matrix to be predicted $X_p$, and $N$ equal to the number of resamples (e.g. 10,000):
for $i$ in $N$:
Then the bootstrap estimate and prediction intervals can be computed directly from that vector (e.g. mean and quantiles of the vector).
So my question is why is step 4 necessary? My intuition is that it has to do with the fact that I'm computing prediction intervals rather than a confidence interval, but I haven't found a good resource on this.
Sources: Slides 12-15 https://www.emse.fr/~roustant/Documents/Bootstrap_Conf_and_Pred_Intervals.pdf
Bootstrap prediction interval (This would be perfect if I happened to be using OLS, but I'm not...)