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Gaussian process regression estimates an unknown function by assigning its values a Gaussian process law, then conditioning on observed values, often with independent Gaussian noise. The covariance function expresses the proposed relation between nearby inputs; the conditional multivariate normal distribution gives the resulting Gaussian process regression posterior. Its conditional mean provides a prediction and its conditional variance quantifies uncertainty under the chosen model. The index set can represent spatial locations or other inputs as well as time.

At the fitted hyperparameters, define $K=k_0(t,t)$, $K_{\ast \ast }=k_0(t^{\ast },t^{\ast })$, and $K_{\ast t}=k_0(t^{\ast },t)$, with the matrices formed by pairwise kernel evaluations. Conditional independence of observation noise and future latent values givesThe residual $f_{\ast }-K_{\ast t}{V}^{-1}y$ has zero covariance with $y$, so the joint normal law makes it independent of $y$. This derives the conditional multivariate normal distribution and the Gaussian process regression posterior:The covariance is a Schur complement and is positive semidefinite. When nonsingular, its joint density is $(2\pi {)}^{-M/2}|{\Sigma }_{\ast }{|}^{-1/2}\exp [-\frac{1}{2}(f_{\ast }-m_{\ast }{)}^T{\Sigma }_{\ast }^{-1}(f_{\ast }-m_{\ast })]$. Repeated phases can make it singular, in which case the displayed normal law is interpreted on its support.

This predicts the true latent light curve, so no future measurement variance is added. For noisy future observations, add their independent noise covariance to ${\Sigma }_{\ast }$. Fixing the hyperparameters omits their estimation uncertainty, as requested; integrating over their posterior would add it. The periodic model extrapolates by phase, even though all prediction times follow the observed times.

At the fitted parameters, let $C=C_{\hat{\theta }}$ and $\mu ={\mu }_{\hat{\theta }}$. For prediction times $t_{\ast }$ defineThe microlensing processes and measurement errors contribute no cross-covariance with the latent quasar light curve. The Gaussian process regression posterior is thereforeThe requested pointwise posterior variances are the diagonal entries of the latter matrix.

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 $t_1$.

Let $K\in {\mathbb{R}}^{n\times n}$ have entries $K_{ij}=\kappa (x_i,x_j)$, let $k_{\ast }\in {\mathbb{R}}^n$ have entries $(k_{\ast }{)}_i=\kappa (x_i,x_{\ast })$, and let $k_{\ast \ast }=\kappa (x_{\ast },x_{\ast })$. The Gaussian process prior and independent Gaussian noise implyApplying the conditional multivariate normal distribution gives the Gaussian process regression posteriorwhere

Set $x_{\ast }=0.5$ in the Gaussian process regression posterior and compute $m_{\ast }$ and $v_{\ast }$ as in part a. Standardization of a normal random variable then giveswhere $\Phi$ is the standard normal cumulative distribution function.