At the fitted parameters, let $C=C_{\widehat\theta}$ and $\mu=\mu_{\widehat\theta}$. For prediction times $t_*$ define
$$
K_{**}=K_f(t_*,t_*),
\qquad
K_{*y}=\begin{pmatrix}K_f(t_*,t)&K_f(t_*,t-\widehat{\Delta t})\end{pmatrix}.
$$
The microlensing processes and measurement errors contribute no cross-covariance with the latent quasar light curve. The <Gaussian process regression posterior> is therefore
$$
\boxed{\mathbb E[f_*\mid y]=c\mathbf1+K_{*y}C^{-1}(y-\mu),}
$$
$$
\boxed{\operatorname{Cov}(f_*\mid y)
=K_{**}-K_{*y}C^{-1}K_{y*}.}
$$
The requested pointwise posterior variances are the diagonal entries of the latter matrix.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 219 / 3 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-219/3/b/solution