Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 219 / 2 / a / Solution

The vector $(y_1,y_2,y_3)$ is Jointly Gaussian. Assuming $R>0$, Gaussian conditional independence givesThus the required condition is $R{R}_{13}=R_{12}{R}_{23}$. Under it, conditioning on $y_1$ supplies no further information after $y_2$, and the Gaussian process regression posterior isBoth the conditional expectation and conditional variance depend only on $y_2,t_2,t_3$; neither contains $y_1$ or $t_1$.