likelihood $L^I({\boldsymbol \sigma}^2, {\boldsymbol \phi}\mid {\mathbf{y}})$: $$\begin{aligned}
    \begin{split}
    L^I({\boldsymbol \sigma}^2, {\boldsymbol \phi}\mid {\mathbf{y}}) &= \int_{\mathbb{R}^{q+1}} L({\boldsymbol \theta}\mid {\mathbf{y}}) \pi^R({\boldsymbol \beta}, {\boldsymbol \gamma}) d ({\boldsymbol \beta}, {\boldsymbol \gamma}) \\
        & \propto \prod_{t=1}^s (\sigma^2_t)^{-(n_t-q_t)/2}|{\mathbf{R}}_t|^{-1/2}|{\mathbf{X}}_t^\top {\mathbf{R}}_t^{-1} {\mathbf{X}}_t|^{-1/2} \exp\left \{-\frac{S^2({\boldsymbol \phi}_t)}{2\sigma^2_t} \right\}.
    \end{split}\end{aligned}$$

\[thm: reference prior\] Consider the group of parameters ${\boldsymbol \theta}=({\boldsymbol \theta}_1, \ldots, {\boldsymbol \theta}_s)$ with ${\boldsymbol \theta}_t=({\boldsymbol \beta}_t, \gamma_{t-1}, \sigma^2_t, {\boldsymbol \phi}_t)$, where $\gamma_0:=0$. For the cokriging model with sampling distribution , the independent reference prior distribution, $\pi^R({\boldsymbol \theta})$, is of the form  with $$\begin{aligned}
    a_t = 1 & \text{ and } \pi^R({\boldsymbol \phi}) \propto \prod_{t=1}^s|I_t^R({\boldsymbol \phi}_t)|^{1/2},\end{aligned}$$ where $I_t^R({\boldsymbol \phi}_t)$ is the Fisher information matrix by fixing all parameters except ${\boldsymbol \theta}_t$: $$\begin{aligned}
I_t^R({\boldsymbol \phi}_t) = 
\begin{pmatrix}
n_t-q_t & \text{tr}({\mathbf{W}}_{t,1})   & \text{tr}({\mathbf{W}}_{t,2})      & \cdots & \text{tr}({\mathbf{W}}_{t,d}) \\
    & \text{tr}({\mathbf{W}}_{t,1}^2) & \text{tr}({\mathbf{W}}_{t,1}{\mathbf{W}}_{t,2}) & \cdots & \text{tr}({\mathbf{W}}_{t,1}{\mathbf{W}}_{t,d}) \\ 
    &                     &          & \ddots    & \vdots  \\
    &                     &         &   & \text{tr}({\mathbf{W}}_{t,d}^2)
\end{pmatrix}_{(d+1)\times (d+1)},\end{aligned}$$ with ${\mathbf{W}}_{t,k}=\dot{{\mathbf{R}}}^k_t {\mathbf{Q}}_t, k=1, \ldots, d$,

See Appendix \[app: reference prior\].

\[thm: Jeffreys prior\] Let ${\boldsymbol \theta}=({\boldsymbol \theta}_1, \ldots, {\boldsymbol \theta}_s)$ be the group of parameters with ${\boldsymbol \theta}_t=({\boldsymbol \beta}_t, \gamma_{t-1}, \sigma^2_t, {\boldsymbol \phi}_t)$, where $\gamma_0:=0$. Suppose that ${\boldsymbol \theta}_t$’s are independent. Then the independent Jeffreys prior, $\pi^{J1}$, obtained by assuming 