  The quantification of uncertainties of computer simulations due to input
parameter uncertainties is paramount to assess a model's credibility. For
computationally expensive simulations, this is often feasible only via
surrogate models that are learned from a small set of simulation samples. The
surrogate models are commonly chosen and deemed trustworthy based on heuristic
measures, and substituted for the simulation in order to approximately
propagate the simulation input uncertainties to the simulation output. In the
process, the contribution of the uncertainties of the surrogate itself to the
simulation output uncertainties are usually neglected. In this work, we
specifically address the case of doubtful surrogate trustworthiness, i.e.
non-negligible surrogate uncertainties. We find that Bayesian probability
theory yields a natural measure of surrogate trustworthiness, and that
surrogate uncertainties can easily be included in simulation output
uncertainties. For a Gaussian likelihood for the simulation data, with unknown
surrogate variance and given a generalized linear surrogate model, the
resulting formulas reduce to simple matrix multiplications. The framework
contains Polynomial Chaos Expansions as a special case, and is easily extended
to Gaussian Process Regression. Additionally, we show a simple way to
implicitly include spatio-temporal correlations. Lastly, we demonstrate a
numerical example where surrogate uncertainties are in part negligible and in
part non-negligible.
