concentrated restricted likelihood to obtain estimates for correlation parameters at each code level. However, whether the choices of priors in [@Qian2008; @Gratiet2013] will lead to good estimates is not discussed. We show that vague proper priors for correlation parameters in [@Qian2008] will lead to an improper posterior. Thus, the usage of such vague priors will not solve but hide the problem, see [@Berger2006] for detailed discussion and references therein. We also show that the concentrated restricted likelihood in [@Gratiet2013] with noninformative priors and informative priors (when chosen to be vague) can have nonrobust estimates in autoregressive cokriging models, where nonrobustness is defined to be the situation where the correlation matrix becomes either singular or identity when correlation parameters go to zero or infinity; see [@Gu2018] for detailed discussions.

This article has two primary objectives that are of interest from computational and theoretical perspectives. The first objective is the derivation of new formulas for predictive distributions of the code output at any level over a new input given code output and correlation parameters. Realizations can be simulated from predictive distributions based upon a set of conditional distributions and the predictive means and predictive variances can be computed exactly at any fidelity 