  It is generally accepted that all models are wrong -- the difficulty is
determining which are useful. Here, a useful model is considered as one that is
capable of combining data and expert knowledge, through an inversion or
calibration process, to adequately characterize the uncertainty in predictions
of interest. This paper derives conditions that specify which simplified models
are useful and how they should be calibrated. To start, the notion of an
optimal simplification is defined. This relates the model simplifications to
the nature of the data and predictions, and determines when a standard
probabilistic calibration scheme is capable of accurately characterizing
uncertainty. Furthermore, two additional conditions are defined for suboptimal
models that determine when the simplifications can be safely ignored. The first
allows a suboptimally simplified model to be used in a way that replicates the
performance of an optimal model. This is achieved through the judicial
selection of a prior term for the calibration process that explicitly includes
the nature of the data, predictions and modelling simplifications. The second
considers the dependency structure between the predictions and the available
data to gain insights into when the simplifications can be overcome by using
the right calibration data. Furthermore, the derived conditions are related to
the commonly used calibration schemes based on Tikhonov and subspace
regularization. To allow concrete insights to be obtained, the analysis is
performed under a linear expansion of the model equations and where the
predictive uncertainty is characterized via second order moments only.
