# Are Gaussian Process learning problems well defined with only a covariance function?

Lets consider a simple derivation of GPs I've seen.

Given a dataset $( X, \vec{y} ) = (\vec{x}_i, y_i)_{i=0}^N$, we wish to find a function $f$ such that $y_i = \vec{w} \cdot \vec{\phi}(\vec{x_i}) + \epsilon_i$ where $\vec{\epsilon} \sim \mathcal{N}\big( 0, \sigma^2 \mathbb{1} \big)$ is the gaussian noise. From various sources, it seems typical to assume $w \sim \mathcal{N}(0, \mathbb{1})$ (sometimes with a non-unit variance).

Then, $\vec{y} \sim \mathcal{N}(0, \Phi \Phi^T + \sigma^2)$ with $\Phi_{ij} = \phi_i(\vec{x}_j)$. This looks great because we don't have any parameters anymore, but this simple computes $Pr(\vec{y} | X)$ and ignores the output of $\vec{w} \Phi$!

I would imagine we would want to add priors for $\phi$ and use $Pr(\vec{y} | \Phi, X) = \mathcal{N}(w \Phi, \sigma^2 \mathbb{1})$ (by inverting the noise distribution) for prediction.

How are we justified in ignoring $\vec{w} \Phi$ and only using $Pr(\vec{y})$? What am I misunderstanding? 