Predictive variance ($\mathrm{Var}(Y_{\ast }\mid \mathcal{D})$)

The variance of a predictive distribution quantifies spread of a future quantity conditional on the observed data and model assumptions. For a noisy future observation $Y_{\ast }=f_{\ast }+{ϵ}_{\ast }$ with fresh independent noise of variance ${\sigma }^2$, it is $\mathrm{Var}(f_{\ast }\mid \mathcal{D})+{\sigma }^2$. It should therefore be distinguished from the latent-function posterior variance.