In [@Gratiet2014], a recursive predictive formula is also derived based on a recursive multifidelity model that is different from what is presented in this article. The difference between the recursive multifidelity model in [@Gratiet2014] and the one presented here is that [@Gratiet2014] represent the high-fidelity code output $y_t(\cdot)$ as a function of the Gaussian process $y_{t-1}(\cdot)$ conditional on the code output ${\mathbf{y}}_{t-1}^{\mathscr{D}} :=\{ {\mathbf{y}}_1, \ldots, {\mathbf{y}}_{t-1} \}$ at design points in the experimental design sets $\{ \mathcal{X}_{\ell}: \ell=1, \ldots, t-1\}$ and model parameters $\{{\boldsymbol \beta}_{t-1}, \gamma_{t-1}, \sigma^2_{t-1}, {\boldsymbol \phi}_{t-1} \}$. Essentially, this recursive multifidelity model models the predictive distribution of $\pi(y_t(\cdot) \mid {\mathbf{y}}_{t-1}^{\mathscr{D}}, {\boldsymbol \beta}_{t-1}, \gamma_{t-1}, \sigma^2_{t-1}, {\boldsymbol \phi}_{t-1})$. This recursive multifidelity model only needs to estimate model parameters and prediction can be made automatically based on the model, however, it is worth noting that this predictive distribution does not take into account uncertainty due to estimation of model parameters $\{ {\boldsymbol \beta}, {\boldsymbol \gamma}, {\boldsymbol \sigma}^2, {\boldsymbol \phi}\}$ and it does not give the predictive formula to predict code output at intermediate levels, $t=1, \ldots, s-1$, where such predictions may be useful to design the experiment with multifidelity codes.

The following corollary highlights the properties of autoregressive cokriging predictors 