Marginalization example - uncertainty

[andrewfowlie]

In this example

https://tinygp.readthedocs.io/en/stable/tutorials/modeling.html#modeling-numpyro

We marginalize hyperparameters with HMC. In the analysis, we plot quantiles of the posterior mean of y found for each state of hyperparameters in the chain.

I'm a bit confused about what the quantiles of the posterior mean are supposed to represent? In my understanding, they show the uncertainty on the mean of y, rather than the uncertainty on y, since they don't account for the (co)variance.

For example, I could know the mean of y exactly. I still wouldn't know y, though, because of the GP has a covariance as well as a mean.