that $({\boldsymbol \beta}_t, \gamma_t)$ and $(\sigma_t^2, {\boldsymbol \phi}_t)$ are a priori independent, and the independent Jeffreys prior, $\pi^{J2}$, are of the form  with $$\begin{aligned}
    a_t = 1 & \text{ and } \pi^{J1}({\boldsymbol \phi}) \propto \prod_{t=1}^s|I_t^J({\boldsymbol \phi}_t)|^{1/2}, \\
    a_t = 1 + q_t/2 & \text{ and } \pi^{J2}({\boldsymbol \phi}) \propto \pi^{J1}({\boldsymbol \phi}) \prod_{t=1}^s |{\mathbf{X}}_t^\top {\mathbf{R}}^{-1}_t {\mathbf{X}}_t|^{1/2} ,\end{aligned}$$ where $$\begin{aligned}
I_t^J({\boldsymbol \phi}_t) = 
\begin{pmatrix}
n_t & \text{tr}({\mathbf{U}}_{t,1})   & \text{tr}({\mathbf{U}}_{t,2})      & \cdots & \text{tr}({\mathbf{U}}_{t,d}) \\
    & \text{tr}({\mathbf{U}}_{t,1}^2) & \text{tr}({\mathbf{U}}_{t,1}{\mathbf{U}}_{t,2}) & \cdots & \text{tr}({\mathbf{U}}_{t,1}{\mathbf{U}}_{t,d}) \\ 
    &                     &          & \ddots    & \vdots  \\
    &                     &         &   & \text{tr}({\mathbf{U}}_{t,d}^2)
\end{pmatrix}_{(d+1)\times (d+1)},\end{aligned}$$ with ${\mathbf{U}}_{t,k}=\dot{{\mathbf{R}}}^k {\mathbf{R}}_t^{-1}, k=1, \ldots, d$,

This result follows directly from Proposition 2.2 in [@Paulo2005].

For the cokriging model with sampling distribution , the independent reference prior $\pi^R$, independent Jeffreys priors $\pi^{J1}, \pi^{J2}$ yield proper posteriors satisfying the condition .

Following the results in [@Paulo2005], it is straightforward to show that $$\begin{aligned}
    0<\int_{\mathbf{R}^d} L^I({\boldsymbol \phi}_t\mid{\mathbf{y}}) \pi({\boldsymbol \phi}_t)\, d{\boldsymbol \phi}_t < \infty,\quad & \text{ for } \quad  \pi({\boldsymbol \phi}_t)=\pi^R({\boldsymbol \phi}_t), \pi^{J1}({\boldsymbol \phi}_t), \pi^{J2}({\boldsymbol \phi}_t),\end{aligned}$$ where $L^I({\boldsymbol \phi}_t\mid{\mathbf{y}}) \propto |{\mathbf{R}}_t|^{-1/2}|{\mathbf{X}}_t^\top {\mathbf{R}}_t^{-1} {\mathbf{X}}_t|^{-1/2} \{S^2({\boldsymbol \phi}_t)\}^{-(({n}_t - q_t)/2+a_t-1)}$, $\pi^R({\boldsymbol \phi}_t) \propto I_t^R({\boldsymbol \phi}_t)$, $\pi^{J1}({\boldsymbol \phi}_t) \propto I_t^J({\boldsymbol \phi}_t)$, $\pi^{J2}({\boldsymbol \phi}_t) \propto I_t^J({\boldsymbol \phi}_t) |{\mathbf{X}}_t^\top {\mathbf{R}}_t^{-1} {\mathbf{X}}_t|^{1/2}$. Then the condition  