  We introduce Bayesian Probability Theory to investigate uncertainty
propagation based on meta-models. We approach the problem from the perspective
of data analysis, with a given (however almost-arbitrary) input probability
distribution and a given "training" set of computer simulations. While proven
mathematically to be the unique consistent probability calculus, the subject of
this paper is not to demonstrate beauty but usefulness. We explicitly list all
propositions and lay open the general structure of any uncertainty propagation
based on meta-models. The former allows rigorous treatment at any stage, while
the latter allows us to quantify the interaction of the surrogate uncertainties
with the usual parameter uncertainties. Additionally, we show a simple way to
implicitly include spatio-temporal correlations. We then apply the framework
jointly to a family of generalized linear meta-model that implicitly includes
Polynomial Chaos Expansions as a special case. While we assume a Gaussian
surrogate-uncertainty, we do not assume a scale for the surrogate uncertainty
to be known, i.e. a Student-t. We end up with semi-analytic formulas for
surrogate uncertainties and uncertainty propagation
