Given data $(\textbf{X,Y})$, a Gaussian Process $\textbf{F}$ that is normally distributed and weights $\textbf{W}_1$ and $\textbf{b}$ also from a distribution, how come is the following distribution

\begin{align} p(\textbf{Y}|\textbf{X}) = \int p(\textbf{Y}|\textbf{F})p(\textbf{F}|\textbf{W}_1,\textbf{b},\textbf{X})p(\textbf{W}_1)p(\textbf{b}) \label{ref1} \end{align}

a predictive distribution?

Because I thought that a predictive distribution is always, given the data $(\textbf{X,Y})$, the propability of the target label $y^*$ given the new data point $x^*$ data, such as

\begin{align*} p(y^*|x^*,\textbf{X,Y}) = \int p(y^*|x^*,\pmb{\omega}) p(\pmb{\omega}|\textbf{X,Y}) \, d \pmb{\omega} \end{align*}

for some parameters $\pmb{\omega}$.

Currently Im investigating this paper, and in it (section 3.1.) I tripped over the above, for me incomprehensible, predictive distribution.

Anyone has a trick up their sleeve and knows by chance whats going on here?

Cheers