Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 207 / 3 / a / Solution

Let $K\in {\mathbb{R}}^{n\times n}$ have entries $K_{ij}=\kappa (x_i,x_j)$, let $k_{\ast }\in {\mathbb{R}}^n$ have entries $(k_{\ast }{)}_i=\kappa (x_i,x_{\ast })$, and let $k_{\ast \ast }=\kappa (x_{\ast },x_{\ast })$. The Gaussian process prior and independent Gaussian noise implyApplying the conditional multivariate normal distribution gives the Gaussian process regression posteriorwhere