Understanding likelihoods for Gaussian Processes What does it mean when we talk about the "Gaussian likelihood" for a Gaussian Process? Is it true to think that the "Gaussian likelihood" only means we factor in a noise term into the covariance function? i.e.
$$cov(x,x') = K(x,x') + \sigma^2$$
In other words, we only add $\sigma^2$ to the diagonal of the covariance matrix. Is my understanding correct? Furthermore, what happens if the variance of the Gaussian likelihood is zero? Does the likelihood become non-Gaussian and we can't use the standard Gaussian Process inference any more?
 A: The terminology around that is confusing.  On its own, a Gaussian process is a prior distribution over a function $f(x)$, $p(f \mid x)$.  The most common (by far!) case you see discussed is when that function additionally has IID Gaussian noise added to it, call it $y$.  $y = f + \epsilon$, where $\epsilon \sim N(0, \sigma^2)$.  Bayes theorem here is
$$
p(f, \sigma^2 \mid x, y) = \frac{p(y \mid f, x, \sigma^2) p(f \mid x) p(\sigma^2)}{p(y \mid x)}
$$
The likelihood, $p(y \mid f, x, \sigma^2)$ is a product of normals, each with mean $f_i$ and variance $\sigma^2$, and $p(f \mid x)$ is a multivariate normal, the GP prior.  
These distributions are conjugate, so $f$ can be integrated out analytically, producing the marginal likelihood.  This marginal likelihood ends up being a multivariate normal whose covariance is, like you had, $K(x, x') + \sigma^2$.  
So your understanding is partially correct.  If the variance of the Gaussian likelihood goes to zero, you absolutely still use GPs.  All that happens is that Bayes theorem above gets a bit simpler.  
This answer may help you also.
