Models are often defined through conditional rather than joint distributions,
but it can be difficult to check whether the conditional distributions are
compatible, i.e. whether there exists a joint probability distribution which
generates them. When they are compatible, a Gibbs sampler can be used to sample
from this joint distribution. When they are not, the Gibbs sampling algorithm
may still be applied, resulting in a "pseudo-Gibbs sampler". We show its
stationary probability distribution to be the optimal compromise between the
conditional distributions, in the sense that it minimizes a mean squared misfit
between them and its own conditional distributions. This allows us to perform
Objective Bayesian analysis of correlation parameters in Kriging models by
using univariate conditional Jeffreys-rule posterior distributions instead of
the widely used multivariate Jeffreys-rule posterior. This strategy makes the
full-Bayesian procedure tractable. Numerical examples show it has near-optimal
frequentist performance in terms of prediction interval coverage
