The role of noise in GPs#

We show the difference between “noisy” and “noiseless” predictions w.r.t. the posterior predictive covariance matrix. When learning a noise model (${\sigma }_n^2>0$, e.g. using a WhiteKernel component in sklearn), then there are two flavors of that covariance matrix. Borrowing from the GPy library’s naming scheme, we have

• predict_noiseless: $\mathrm{cov}({\mathit{f}}_{\ast })$

• predict: $\mathrm{cov}({\mathit{f}}_{\ast })+{\sigma }_n^2\mathbf{I}$

where $\mathrm{cov}({\mathit{f}}_{\ast })$ is the posterior predictive covariance matrix ([RW06] eq. 2.24). These lead to different uncertainty estimates and posterior samples, as will be shown.

Contents