From (2.9) in Chapter 2 of the book by Rasmussen, prediction using Gaussian processes in the weight-space view is
$$\begin{aligned}p\left(f_{*} | \mathbf{x}_{*}, X, \mathbf{y}\right) &= \int p\left(f_{*} | \mathbf{x}_{*}, \mathbf{w}\right) p(\mathbf{w} | X, \mathbf{y}) d \mathbf{w} \\ &= \mathcal{N}\big(\frac{1}{\sigma_n^2}\mathbf{x_*^T}A^{-1}X\mathbf{y}, \mathbf{x_*^T}A^{-1}\mathbf{x_*}\big) \end{aligned}$$
where $A = \frac{1}{\sigma_n^2}XX^T + \Sigma_p^{-1}$, with $\Sigma_p$ being the covariance matrix of prior on $\mathbf{w}$.
I do not understand the first term of the integrand.
Assuming (this might be wrong) $p(f_*|\mathbf{x_*, w})=\mathcal{N}(\mathbf{w^Tx^*},\sigma_f^2)$, the integral yields $$\mathcal{N}\big(\frac{1}{\sigma_n^2}\mathbf{x_*^T}A^{-1}X\mathbf{y}, \sigma_f^2 + \mathbf{x_*^T}A^{-1}\mathbf{x_*}\big)$$
This matches the expression given in the book if $\sigma_f=0$. Does this mean $p(f_*|\mathbf{x_*, w})$ is a Dirac Delta distribution, centered at $\mathbf{w^Tx^*}$?
Additionally, slide 30 here mentions
$$P(f_*|\mathbf{y},X) = \int d\theta P(f_*|\mathbf{y},X,\theta) P(\theta|\mathbf{y},X)$$ The first term in the integrand is tractable.
What does the last statement mean?
I had earlier asked this on CrossValidated, but did not receive any response. Deleted from there.
Questions similar to this on CV: here. But they do not answer my question.