  Posterior predictive p-values (ppps) have become popular tools for Bayesian
model assessment, being general-purpose and easy to use. However,
interpretation can be difficult because their distribution is not uniform under
the hypothesis that the model did generate the data. Calibrated ppps (cppps)
can be obtained via a bootstrap-like procedure, yet remain unavailable in
practice due to high computational cost. This paper introduces methods to
enable efficient approximation of cppps and their uncertainty for fast model
assessment. We first investigate the computational trade-off between the number
of calibration replicates and the number of MCMC samples per replicate.
Provided that the MCMC chain from the real data has converged, using short MCMC
chains per calibration replicate can save significant computation time compared
to naive implementations, without significant loss in accuracy. We propose
different variance estimators for the cppp approximation, which can be used to
confirm quickly the lack of evidence against model misspecification. As
variance estimation uses effective sample sizes of many short MCMC chains, we
show these can be approximated well from the real-data MCMC chain. The
procedure for cppp is implemented in NIMBLE, a flexible framework for
hierarchical modeling that supports many models and discrepancy measures.
