Past exam of the advanced topics in statistical machine learning course of the University of Oxford / 2025 / 3 / b / iii / Solution

The conditional multivariate normal distribution giveswhereFor a direct justification, form $R=f_{\ast }-k_{\ast }^T{C}^{-1}\mathbf{y}$. It is jointly Gaussian with $\mathbf{y}$, has zero mean, and satisfiesThe independence of uncorrelated jointly normal variables therefore makes $R$ independent of $\mathbf{y}$. Conditional on $\mathbf{y}$, only the displayed affine shift remains, proving the result. This is the latent-function posterior variance; a new observation $y_{\ast }=f_{\ast }+\sigma {ϵ}_{\ast }$ with fresh independent noise has predictive variance $v_n(x_{\ast })+{\sigma }^2$.