level in a computationally efficient way. The new closed-form predictive formulas will take into account uncertainties due to the estimation of the location and scale parameters. The second objective is the development of objective priors. The objective priors can be used as default priors when elicitation of prior information is challenging. It also enables more accurate uncertainty estimation in the predictive distribution than the typical maximum-likelihood based approaches with commonly-used noninformative priors.

Section \[sec: univariate model\] gives the general assumptions and reviews the autoregressive cokriging models. In Section \[sec: prediction in univariate model\], new closed-form expressions are derived for predictive distributions at all code levels conditioned on all code output and correlation parameters. The $s$-level cokriging model turns out to have the same computational cost as $s$ independent kriging models in both parameter estimation and prediction. Section \[sec: objective Bayes\] begins with discussions on commonly-chosen noninformative priors and proves that the resulting posteriors are improper with such noninformative priors. The objective priors including independent reference priors and independent Jeffreys priors are then developed and are shown to yield proper posteriors. Section \[sec: numerical demonstration\] gives several numerical examples. Section \[sec: discussion\] is concluded with further discussions.

The Autoregressive Cokriging Model 