7.3.1. Prediction values
Once both the posterior parameter values and covariances are estimated, it is possible to make a prediction for a dataset not used in the estimation. The central value of the prediction is easy to get, as with any other methods presented in this documentation, since one knows the model and can use the newly estimated posterior mean values of the parameters.
The novel aspect is that a variance can also be estimated for the predicted mean using the posterior covariance matrix of the parameters, \( \Sigma^{post}_{\theta} \), already introduced in Equation 7.8. This variance represents the uncertainty in each new predicted point due to parameter uncertainty, and it is contained in the covariance matrix \(\Sigma^{pred}_{\theta} \) of dimension \((q,q)\), where \(q\) is the sample size under consideration. To obtain the estimate, one needs the new design matrix \( H_{pred} = [h(\mathbf{x}_1), \ldots, h(\mathbf{x}_q)]^T \in M_{q,p}(\mathbb{R})\) which then leads to
