Predictive distribution

A probability law for a future observation conditional on currently available information. A posterior predictive distribution integrates the observation model over the posterior distribution of unknown parameters; a plug-in predictive law instead fixes an estimate of those parameters.

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.