  The Bayesian Synthetic Likelihood (BSL) method is a widely-used tool for
likelihood-free Bayesian inference. This method assumes that some summary
statistics are normally distributed, which can be incorrect in many
applications. We propose a transformation, called the Wasserstein
Gaussianization transformation, that uses a Wasserstein gradient flow to
approximately transform the distribution of the summary statistics into a
Gaussian distribution. BSL also implicitly requires compatibility between
simulated summary statistics under the working model and the observed summary
statistics. A robust BSL variant which achieves this has been developed in the
recent literature. We combine the Wasserstein Gaussianization transformation
with robust BSL, and an efficient Variational Bayes procedure for posterior
approximation, to develop a highly efficient and reliable approximate Bayesian
inference method for likelihood-free problems.
