the correlation parameters in the higher level very well, since in practice the higher level code is too expensive to get sufficient number of runs that can be used to obtain fairly good parameter estimates.

Objective Bayesian Analysis {#sec: objective Bayes}
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In Uncertainty Quantification, Bayesian analysis of cokriging models has been focused on using conjugate priors and noninformative priors [@Kennedy2000; @Qian2008; @Gratiet2013]. In this section, we focus on objective Bayesian analysis of the cokriging model, since objective priors can be used as default priors for Bayesian analysis [@Berger2006], and have been often used in Gaussian process modeling [@Berger2001; @Paulo2005; @Gu2018].

Commonly-used Improper Priors
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A Gaussian process has commonly-used priors; see [@Berger2001] for detailed discussions. Following this convention, the following priors will be referred to as commonly-used priors for parameters in autoregressive cokriging models. We consider the improper prior density for ${\boldsymbol \theta}:=\{{\boldsymbol \beta}, {\boldsymbol \gamma}, {\boldsymbol \sigma}^2, {\boldsymbol \phi}\} \in \Omega=\mathbb{R}^{sp} \times \mathbb{R}^s \times (0, \infty)^s\times (0, \infty)^{sd}$ of the form $$\begin{aligned}
\label{eqn: general prior}
\begin{split}
    \pi({\boldsymbol \beta},{\boldsymbol \gamma},{\boldsymbol \sigma}^{2},{\boldsymbol \phi})
    & \propto \frac{\pi({\boldsymbol \phi})}{\prod_{t=1}^s(\sigma^2_t)^{a_t}}, a_t \in \mathbb{R},
            \end{split}\end{aligned}$$ for various choices of $\pi({\boldsymbol \phi})$ and $a_t$. Although Kennedy and O’Hagan [@Kennedy2000] did not assume a prior for 