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We analyse the behaviour of the synthetic likelihood (SL) method when themodel generating the simulated data differs from the actual data generatingprocess. One of the most common methods to obtain SL-based inferences is viathe Bayesian posterior distribution, with this method often referred to asBayesian synthetic likelihood (BSL). We demonstrate that when the model ismisspecified, the BSL posterior can be poorly behaved, placing significantposterior mass on values of the model parameters that do not represent the truefeatures observed in the data. Theoretical results demonstrate that inmisspecified models the BSL posterior can display a wide range of behavioursdepending on the level of model misspecification, including beingasymptotically non-Gaussian. Our results suggest that a recently proposedrobust BSL approach can ameliorate this behavior and deliver reliable posteriorinference under model misspecification. We document all theoretical resultsusing a simple running example.

Synthetic Likelihood in Misspecified Models: Consequences and Corrections