 Solution

Condition throughout on the training inputs and $x_{\ast }$. Interpret $\theta$ here as the fixed covariance function and noise hyperparameters. Assume observation errors ${ϵ}_i$ are independent standard normal random variables, independent of the Gaussian process $f$, and writeThe independent-error assumption is the standard Gaussian process regression interpretation of the observation model; the single-observation marginal noise law alone would not specify cross-observation noise correlations. We take ${\sigma }^2>0$ so that $C$ is a positive-definite matrix. More generally the inverse formulas below apply whenever the indicated noisy covariance matrices are invertible.

The Gaussian process prior makes $(f_{\ast },\mathbf{f})$ a Gaussian random vector. Adding the independent Gaussian noise vector to $\mathbf{f}$ preserves a multivariate normal distribution. Its mean is zero, its observation covariance matrix is $C$, and $\mathrm{Cov}(f_{\ast },y_i)=k(x_{\ast },x_i)$ because the noise is independent of $f_{\ast }$. ThereforeA degenerate latent variance is allowed; the displayed multivariate normal distribution then denotes a Gaussian law even if it has no full-dimensional probability density.