Gaussian process - posterior covariance matrix

Question

In Gaussian Process, what is the posterior covariance matrix?

Let's use terminology from Rasmussen & Williams 2006 (Gaussian processes for machine learning): Say we have Gaussian process $f \sim \mathcal{N}(\mu, K)$, and a likelihood function and a matrix of its second derivatives: $W = -\frac{\partial^2}{\partial^2 f} p(y|f)$.

Then $K$ is the "prior" covariance matrix. What is then the covariance matrix of the posterior estimates of $f$?

Answer 1

Posterior covariance matrix $cov(f|X,y)$ is, under Gaussian approximatin, equal to $(K^{−1} +W)^{−1}$, as apparent from equation $(3.20)$ and the explanations around $(3.24)$ in Rasmussen & Williams 2006 (Gaussian processes for machine learning).