The vector $(y_1,y_2,y_3)$ is <Jointly Gaussian>. Assuming $R>0$, <Gaussian conditional independence> gives
$$
y_3\mathbin\perp y_1\mid y_2
\quad\Longleftrightarrow\quad
\operatorname{Cov}(y_1,y_3\mid y_2)
=R_{13}-\frac{R_{12}R_{23}}R=0.
$$
Thus the required condition is $RR_{13}=R_{12}R_{23}$. Under it, conditioning on $y_1$ supplies no further information after $y_2$, and the <Gaussian process regression posterior> is
$$
y_3\mid y_2,y_1
\sim N\!\left(
\mu+\frac{R_{23}}R(y_2-\mu),
R-\frac{R_{23}^2}R
\right).
$$
Both the <conditional expectation> and <conditional variance> depend only on $y_2,t_2,t_3$; neither contains $y_1$ or $t_1$.

Solution

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

{c}

For a jointly Gaussian vector, conditional independence is equivalent to vanishing conditional cross-covariance. Equivalently, two coordinates are conditionally independent given all the others exactly when their entry in the <precision matrix> is zero.

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