  Bayesian synthetic likelihood (BSL) is now an established method for
conducting approximate Bayesian inference in models where, due to the
intractability of the likelihood function, exact Bayesian approaches are either
infeasible or computationally too demanding. Implicit in the application of BSL
is the assumption that the data generating process (DGP) can produce simulated
summary statistics that capture the behaviour of the observed summary
statistics. We demonstrate that if this compatibility between the actual and
assumed DGP is not satisfied, i.e., if the model is misspecified, BSL can yield
unreliable parameter inference. To circumvent this issue, we propose a new BSL
approach that can detect the presence of model misspecification, and
simultaneously deliver useful inferences even under significant model
misspecification. Two simulated and two real data examples demonstrate the
performance of this new approach to BSL, and document its superior accuracy
over standard BSL when the assumed model is misspecified.
