  Decisions based partly or solely on predictions from probabilistic models may
be sensitive to model misspecification. Statisticians are taught from an early
stage that "all models are wrong", but little formal guidance exists on how to
assess the impact of model approximation on decision making, or how to proceed
when optimal actions appear sensitive to model fidelity. This article presents
an overview of recent developments across different disciplines to address
this. We review diagnostic techniques, including graphical approaches and
summary statistics, to help highlight decisions made through minimised expected
loss that are sensitive to model misspecification. We then consider formal
methods for decision making under model misspecification by quantifying
stability of optimal actions to perturbations to the model within a
neighbourhood of model space. This neighbourhood is defined in either one of
two ways. Firstly, in a strong sense via an information (Kullback-Leibler)
divergence around the approximating model. Or using a nonparametric model
extension, again centred at the approximating model, in order to `average out'
over possible misspecifications. This is presented in the context of recent
work in the robust control, macroeconomics and financial mathematics
literature. We adopt a Bayesian approach throughout although the methods are
agnostic to this position.
