a convenient way to exactly compute the predictive mean and predictive variance in the predictive distribution $\pi({\mathbf{y}}({\mathbf{x}}_0) \mid {\mathbf{y}}, {\boldsymbol \phi})$, we define the cokriging predictor and cokriging variance at each code level.

Let $\hat{{\mathbf{y}}}({\mathbf{x}}_0)=(\hat{y}_1({\mathbf{x}}_0), \ldots, \hat{y}_s({\mathbf{x}}_0))^\top$ be a vector of predictive means with $\hat{y}_t({\mathbf{x}}_0):=E[y_t({\mathbf{x}}_0)\mid {\mathbf{y}}, {\boldsymbol \phi}]$ and $\hat{{\mathbf{v}}}({\mathbf{x}}_0)=(\hat{v}_1({\mathbf{x}}_0),\ldots, \hat{v}_s({\mathbf{x}}_0))^\top$ be a vector of predictive variances with $\hat{v}_t({\mathbf{x}}_0):=Var[y_t({\mathbf{x}}_0)\mid {\mathbf{y}}, {\boldsymbol \phi}]$. In what follows, $\hat{{\mathbf{y}}}({\mathbf{x}}_0)$ is called the cokriging predictor and $\hat{{\mathbf{v}}}({\mathbf{x}}_0)$ is called the cokriging variance for all levels of code at new input ${\mathbf{x}}_0$.

\[thm: predictive mean and variance\] Suppose that $n_t-q_t>2$ such that the $t$-distributions in have valid variances. Then the cokriging predictor and cokriging variance at code level $t$ are given by $$\begin{aligned}
 \label{eqn: cokriging predictor}
    \begin{split}
    \hat{y}_t({\mathbf{x}}_0) &= {\mathbf{f}}_t^\top({\mathbf{x}}_0) \hat{{\mathbf{b}}}_t + {\mathbf{r}}_t^\top({\mathbf{x}}_0) {\mathbf{R}}_t^{-1}({\mathbf{y}}_t - {\mathbf{X}}_t \hat{{\mathbf{b}}}_t), \\
    \hat{v}_t({\mathbf{x}}_0) &= \hat{\gamma}_{t-1}^2\hat{v}_{t-1}({\mathbf{x}}_0) + \frac{n_t-q_t}{n_t-q_t-2} \hat{\sigma}^2_t \{r({\mathbf{x}}_0, {\mathbf{x}}_0|{\boldsymbol \phi}_t) - {\mathbf{r}}_t^\top({\mathbf{x}}_0) {\mathbf{R}}_t^{-1} {\mathbf{r}}_t({\mathbf{x}}_0) + \kappa_t \},
    \end{split}\end{aligned}$$ where ${\mathbf{f}}_1({\mathbf{x}}_0)={\mathbf{h}}_1({\mathbf{x}}_0)$, ${\mathbf{f}}_t({\mathbf{x}}_0)=[{\mathbf{h}}_t({\mathbf{x}}_0)^\top, \hat{y}_{t-1}^\top({\mathbf{x}}_0)]^\top$ for $t>1$, $\hat{v}_0:=0$ and $$\begin{aligned}
    \kappa_t &=  [{\mathbf{f}}_{t}({\mathbf{x}}_0)-{\mathbf{X}}_{t}^{\top}{\mathbf{R}}_{t}^{-1}{\mathbf{r}}_t({\mathbf{x}}_0)]^{\top}({\mathbf{X}}_{t}^{\top}{\mathbf{R}}_{t}^{-1}{\mathbf{X}}_{t})^{-1}[{\mathbf{f}}_{t}({\mathbf{x}}_0)-{\mathbf{X}}_{t}^{\top}{\mathbf{R}}_{t}^{-1}{\mathbf{r}}_t({\mathbf{x}}_0)] \\
    & + \hat{v}_{t-1}({\mathbf{x}}_0) \{ y_{t-1}^\top({{\cal X}}_t) {\mathbf{Q}}^H_t y_{t-1}({{\cal X}}_t) \}^{-1}.\end{aligned}$$ with ${\mathbf{Q}}^H_t = {\mathbf{R}}_t^{-1} - {\mathbf{R}}_t^{-1} {\mathbf{H}}_t ({\mathbf{H}}_t^\top {\mathbf{R}}_t^{-1} {\mathbf{H}}_t)^{-1} {\mathbf{H}}_t^\top {\mathbf{R}}_t^{-1}$.

See Appendix \[app: predictive mean and variance\].

Theorem \[thm: predictive mean and variance\] shows that 