  We analyse the behaviour of the synthetic likelihood (SL) method when the
model generating the simulated data differs from the actual data generating
process. One of the most common methods to obtain SL-based inferences is via
the Bayesian posterior distribution, with this method often referred to as
Bayesian synthetic likelihood (BSL). We demonstrate that when the model is
misspecified, the BSL posterior can be poorly behaved, placing significant
posterior mass on values of the model parameters that do not represent the true
features observed in the data. Theoretical results demonstrate that in
misspecified models the BSL posterior can display a wide range of behaviours
depending on the level of model misspecification, including being
asymptotically non-Gaussian. Our results suggest that a recently proposed
robust BSL approach can ameliorate this behavior and deliver reliable posterior
inference under model misspecification. We document all theoretical results
using a simple running example.
