I have been trying to figure out how to get the marginal likelihood of a GP model. I am working on a regression problem, where my target is $y$ and my inputs are denoted by $x$. The model is $y_i=f(x_i)+\epsilon$, where $\epsilon \sim N(0,\sigma^2)$
Therefore, the marginal likelihood:
$p(y|x) = \int p(y|f,x)p(f|x)df$, where
$p(y|f,x)\sim N(f,\sigma^2I)$ and
I know that the result should be $N(0,K+\sigma^2I)$. However, I am not sure why this is true. If anyone can recommend where I can find the proof or give me a hint I would really appreciate it.