 Gaussian-process marginal likelihood

Let latent values $f$ have a multivariate normal distribution $N(m,K)$, and observe $y=f+\epsilon$ with independent Gaussian noise $\epsilon \sim N(0,E)$. Integrating over $f$ gives $y\sim N(m,C)$, where $C=K+E$. If $C$ is positive-definite, its log-marginal likelihood isThe first term measures the residual under the proposed covariance matrix; the determinant term accounts for the volume of the proposed observation law. Both matter when comparing covariance function parameters. With a single observation, $m=0$ and $C=v>0$, this reduces to $-y^2/(2v)-\log (v)/2-\log (2\pi )/2$.^[ref]

A positive-definite observation-noise covariance matrix $E$ makes $C$ positive-definite even when $K$ is singular; mutually independent errors with strictly positive variances are one sufficient case. If $C$ is singular, a Gaussian observation law still exists but has no ordinary density on the full observation space; the inverse-and-determinant formula above is then not valid. This is a likelihood for kernel and mean parameters after latent-value integration, rather than an assertion that estimated parameters have no further uncertainty.