Let $K\in\mathbb R^{n\times n}$ have entries $K_{ij}=\kappa(x_i,x_j)$, let $k_*\in\mathbb R^n$ have entries $(k_*)_i=\kappa(x_i,x_*)$, and let $k_{**}=\kappa(x_*,x_*)$. The <Gaussian process> prior and independent <Gaussian noise> imply
$$
\begin{pmatrix}f(x_*)\\y\end{pmatrix}
\sim N\!\left[
\begin{pmatrix}0\\0_n\end{pmatrix},
\begin{pmatrix}
k_{**}&k_*^T\\
k_*&K+\sigma^2I_n
\end{pmatrix}
\right].
$$
Applying the <conditional multivariate normal distribution> gives the <Gaussian process regression posterior>
$$
f(x_*)\mid y,X,x_*
\sim N(m_*,v_*),
$$
where
$$
m_*=k_*^T(K+\sigma^2I_n)^{-1}y,
\qquad
v_*=k_{**}-k_*^T(K+\sigma^2I_n)^{-1}k_*.
$$

Solution

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

Set $x_*=0.5$ in the <Gaussian process regression posterior> and compute $m_*$ and $v_*$ as in part a. <Standardization of a normal random variable> then gives
$$
\mathbb P\bigl(f(0.5)<1.5\mid y,X\bigr)
=\Phi\!\left(\frac{1.5-m_*}{\sqrt{v_*}}\right),
$$
where $\Phi$ is the <standard normal cumulative distribution function>.

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

{c}

For a zero-mean <Gaussian process> observed with independent <Gaussian noise>, the latent value at a new input is conditionally normal. Its mean is the kernel cross-covariance times the noisy kernel-matrix inverse times the observations, and its variance is the prior variance minus the corresponding quadratic form.

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