I have an assignment and I'm kinda confused with the terminology of prior in the context of Gaussian processes.
Let $f$ be the function of interest. The matrix $\textbf{X}$ is also a set of $N$ $M$-dimensional input vectors.
According to the description I have been provided, we can formulate the prior over the output of the function f using a Gaussian Process in the following way:
$p(f|\textbf{X}, \textbf{θ}) = Ν(\textbf{0}, k(\textbf{X}, \textbf{X}))$
where $θ$ is the hyperparameters of the kernel $k$.
What troubles me is that this has been called a prior but it depends on the data $\textbf{X}$.
Shouldn't the prior be independent on the data, expressing our belief about some property of the function through the kernel?
Is the notation/terminology used wrong or is there something I haven't understood in the notation?