I was performing some standard model checking and realized strange outputs when recreating plots for posterior predictive checks. My question closely relates to this one; however, I am wondering how this extends into prediction. I created a gist that provides all of the details.

Basically, I see that the distribution of values from a single draw align nicely with the distribution of my observed data. However, the distribution of the mean of the draws does not.

Using `plot_ppc`

from `arviz`

:

Same thing but by hand:

The dark blue lines in mine are the same as the blue lines in the `arviz`

version. The baby blue in mine is the same as the orange in the `arviz`

version. The KDE of all the samples is not the same as the KDE of the mean of the samples. Similarly, red in mine is the same as black in `arviz`

. (probably should have just matched colorsâ€¦)

In the linked post @OriolAbril points out how this fits within a Bayesian context, but my question is how this extends into prediction. For example, in the example I provided, we want to make some predictions on a test set. I completely understand the point that a main advantage of Bayesian methods is that we obtain a distribution instead of just a point estimate. However, in some cases we may need to use a point estimate as our prediction for use in real-world applications (also I get that the advantage is that we can choose what that point estimate is, e.g., MAP, median, â€¦).

Does taking the mean of our predictions (e.g., over `chain`

and `draw`

) have the same impact that it does when we are performing posterior predictive checks? Given that predictions are basically posterior checks on new observations, I donâ€™t see how this would not be something to consider here.

Sorry if this is very basic, but I donâ€™t see quite as much discussion around predictions.

Thanks.