  Bayesian methods have been very successful in quantifying uncertainty in
physics-based problems in parameter estimation and prediction. In these cases,
physical measurements y are modeled as the best fit of a physics-based model
$\eta(\theta)$ where $\theta$ denotes the uncertain, best input setting. Hence
the statistical model is of the form $y = \eta(\theta) + \epsilon$, where
$\epsilon$ accounts for measurement, and possibly other error sources. When
non-linearity is present in $\eta(\cdot)$, the resulting posterior distribution
for the unknown parameters in the Bayesian formulation is typically complex and
non-standard, requiring computationally demanding computational approaches such
as Markov chain Monte Carlo (MCMC) to produce multivariate draws from the
posterior. While quite generally applicable, MCMC requires thousands, or even
millions of evaluations of the physics model $\eta(\cdot)$. This is problematic
if the model takes hours or days to evaluate. To overcome this computational
bottleneck, we present an approach adapted from Bayesian model calibration.
This approach combines output from an ensemble of computational model runs with
physical measurements, within a statistical formulation, to carry out
inference. A key component of this approach is a statistical response surface,
or emulator, estimated from the ensemble of model runs. We demonstrate this
approach with a case study in estimating parameters for a density functional
theory (DFT) model, using experimental mass/binding energy measurements from a
collection of atomic nuclei. We also demonstrate how this approach produces
uncertainties in predictions for recent mass measurements obtained at Argonne
National Laboratory (ANL).
