& \propto \prod_{t=1}^s |{{\mathbf{R}}}_t|^{-1/2} |{{\mathbf{X}}}_t^\top {{\mathbf{R}}}_t^{-1} {{\mathbf{X}}}_t|^{-1/2} \{S^2({\boldsymbol \phi}_t)\}^{-(({n}_t - q_t)/2+a_t-1)}, 
\end{split}\end{aligned}$$ where ${{\mathbf{X}}}_1={{\mathbf{H}}}_1$ and ${{\mathbf{X}}}_t=[{{\mathbf{H}}}_t, {W}_{t-1}]$ with ${W}_{t-1}:=y_{t-1}({{{\cal X}}}_t)$ for $t=2, \ldots, s$. $S^2({\boldsymbol \phi}_t):={\mathbf{y}}_{t}^\top {\mathbf{Q}}_t {\mathbf{y}}_{t}$ with ${\mathbf{Q}}_t={\mathbf{R}}_t^{-1}{\mathbf{P}}$ and ${\mathbf{P}}:=\mathbf{I} - {\mathbf{X}}_t ({\mathbf{X}}_t^\top {\mathbf{R}}_t^{-1} {\mathbf{X}}_t)^{-1} {\mathbf{X}}_t^\top {\mathbf{R}}_t^{-1}$. It immediately follows that the posterior distribution of $\{{\boldsymbol \beta}, {\boldsymbol \gamma}, {\boldsymbol \sigma}^2, {\boldsymbol \phi}\}$ is proper if and only if $$\begin{aligned}
 \label{eqn: condition for properiety}
    0< \int_{(0, \infty)^{sd}} L^I({\boldsymbol \phi}\mid {\mathbf{y}}) \pi({\boldsymbol \phi}) d {\boldsymbol \phi}< \infty. \end{aligned}$$ The following lemma gives the behavior of the integrated likelihood at zero and at infinity.

\[lem: behavior of integrated likelihood\] Note that ${\boldsymbol \phi}_t=(\phi_{t,1}, \ldots, \phi_{t,d})^\top$. Let $\mathcal{C}({\mathbf{X}}_t)$ be the column space of ${\mathbf{X}}_t$. For the cokriging model with sampling distribution  and prior distribution , under mild assumptions, we have

1.  If $\exists \ell$ such that $\phi_{t,\ell} \to 0$ for at least one $t$, the integrated likelihood exists and is greater than zero.

2.  If $\phi_{t,\ell} \to \infty$ for all $\ell$ and $t$, the integrated likelihood satisfies $$\begin{aligned}
            L^I({\boldsymbol \phi}\mid {\mathbf{y}}) =             O\left( \prod_{t=1}^s \left(\sum_{\ell=1}^d \nu_{t,\ell}(\phi_{t,\ell} )\right)^{a_t-c_t} \right),
        \end{aligned}$$

    where $c_t:= 1_{\{\mathbf{1}\in \mathcal{C}({\mathbf{X}}_t)\}} + \frac{1}{2} 1_{\{\mathbf{1}\notin \mathcal{C}({\mathbf{X}}_t)\}}$ with $1_{\{\mathbf{1}\in \mathcal{C}({\mathbf{X}}_t)\}}$ being 1 if the vector $\mathbf{1}$ is in $\mathcal{C}({\mathbf{X}}_t)$ and zero 