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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.

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