Bayesian Surrogate Analysis and Uncertainty Propagation with Explicit Surrogate Uncertainties and Implicit Spatio-temporal Correlations

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