Robust Approximate Bayesian Inference with Synthetic Likelihood

Abstract

Bayesian synthetic likelihood (BSL) is now a well-established method for conducting approximate Bayesian inference in complex models where exact Bayesian approaches are either infeasible, or computationally demanding, due to the intractability of likelihood function. Similar to other approximate Bayesian methods, such as the method of approximate Bayesian computation, implicit in the application of BSL is the maintained assumption that the data generating process (DGP) can generate simulated summary statistics that capture the behaviour of the observed summary statistics. This notion of model compatibility with the observed summaries is critical for the performance of BSL and its variants. We demonstrate through several examples that if the assumed DGP differs from the true DGP, model compatibility may no longer be satisfied and BSL can give unreliable inferences. To circumvent the issue of incompatibility between the observed and simulated summary statistics, we propose two robust versions of BSL that can deliver reliable performance regardless of whether or not the assumed DGP can generate simulated summary statistics that mimic the behavior of the observed summaries. Simulation results and two empirical examples demonstrate the performance of this robust approach to BSL, and its superiority over standard BSL when model compatibility is not in evidence.