In the chapter 2 of [Gaussian Process], equations (2.22-2.24) gives the predictive equations for Gaussian process regression, shown as follows. My question is how to derive f|X,y. It seems that the book does not give the derivation process for f|X,y, the posterior distribution of f given existed observation data.
The proof given in these notes is fairly clear. The approach used is to split the joint distribution of the targets $\mathbf{x}$ and observations $\mathbf{y}$ into two parts representing the marginal distribution $P(\mathbf{y})$ and the conditional distribution $P(\mathbf{x} | \mathbf{y})$. As both of these are Gaussian it's then possible to extract their parameters. It involves quite a bit of matrix algebra, but nothing especially complicated.