  Bayesian synthetic likelihood (BSL) is now a well established method for
performing approximate Bayesian parameter estimation for simulation-based
models that do not possess a tractable likelihood function. BSL approximates an
intractable likelihood function of a carefully chosen summary statistic at a
parameter value with a multivariate normal distribution. The mean and
covariance matrix of this normal distribution are estimated from independent
simulations of the model. Due to the parametric assumption implicit in BSL, it
can be preferred to its non-parametric competitor, approximate Bayesian
computation, in certain applications where a high-dimensional summary statistic
is of interest. However, despite several successful applications of BSL, its
widespread use in scientific fields may be hindered by the strong normality
assumption. In this paper, we develop a semi-parametric approach to relax this
assumption to an extent and maintain the computational advantages of BSL
without any additional tuning. We test our new method, semiBSL, on several
challenging examples involving simulated and real data and demonstrate that
semiBSL can be significantly more robust than BSL and another approach in the
literature.
