In this rejoinder we discuss the following aspects of our approach to model discrepancy: the interpretations of the two populations and adventitious error, the choice of inverse Wishart distribution, the perceived danger of justifying a model with bad fit, the relationship among our new approach, Chen’s (J R Stat Soc Ser B, 41:235–248, 1979) approach and the existing RMSEA-based approach, and the Pitman drift assumption.

Wu and Browne (Psychometrika, 79, 2015) have proposed an innovative approach to modeling discrepancy between a covariance structure model and the population that the model is intended to represent. Their contribution is related to ongoing developments in the field of Uncertainty Quantification (UQ) on modeling and quantifying effects of model discrepancy. We provide an overview of basic principles of UQ and some relevant developments and we examine the Wu–Browne work in that context. We view the Wu–Browne contribution as a seminal development providing a foundation for further work on the critical problem of model discrepancy in statistical modeling in psychological research.

Atmospheric aerosols influence the Earth’s climate, primarily by affecting cloud formation and scattering visible radiation. However, aerosol-related physical processes in climate simulations are highly uncertain. Constraining these processes could help improve model-based climate predictions. We propose a scalable statistical framework for constraining the parameters of expensive climate models by comparing model outputs with observations. Using the C3.AI Suite, a cloud computing platform, we use a perturbed parameter ensemble of the UKESM1 climate model to efficiently train a surrogate model. A method for estimating a data-driven model discrepancy term is described. The strict bounds method is applied to quantify parametric uncertainty in a principled way. We demonstrate the scalability of this framework with 2 weeks’ worth of simulated aerosol optical depth data over the South Atlantic and Central African region, written from the model every 3 hr and matched in time to twice-daily MODIS satellite observations. When constraining the model using real satellite observations, we establish constraints on combinations of two model parameters using much higher time-resolution outputs from the climate model than previous studies. This result suggests that within the limits imposed by an imperfect climate model, potentially very powerful constraints may be achieved when our framework is scaled to the analysis of more observations and for longer time periods.