  Synthetic likelihood is an attractive approach to likelihood-free inference
when an approximately Gaussian summary statistic for the data, informative for
inference about the parameters, is available. The synthetic likelihood method
derives an approximate likelihood function from a plug-in normal density
estimate for the summary statistic, with plug-in mean and covariance matrix
obtained by Monte Carlo simulation from the model. In this article, we develop
alternatives to Markov chain Monte Carlo implementations of Bayesian synthetic
likelihoods with reduced computational overheads. Our approach uses stochastic
gradient variational inference methods for posterior approximation in the
synthetic likelihood context, employing unbiased estimates of the log
likelihood. We compare the new method with a related likelihood free
variational inference technique in the literature, while at the same time
improving the implementation of that approach in a number of ways. These new
algorithms are feasible to implement in situations which are challenging for
conventional approximate Bayesian computation (ABC) methods, in terms of the
dimensionality of the parameter and summary statistic.
