  Implementing Bayesian inference is often computationally challenging in
applications involving complex models, and sometimes calculating the likelihood
itself is difficult. Synthetic likelihood is one approach for carrying out
inference when the likelihood is intractable, but it is straightforward to
simulate from the model. The method constructs an approximate likelihood by
taking a vector summary statistic as being multivariate normal, with the
unknown mean and covariance matrix estimated by simulation for any given
parameter value. Our article makes three contributions. The first shows that if
the summary statistic satisfies a central limit theorem, then the synthetic
likelihood posterior is asymptotically normal and yields credible sets with the
correct level of frequentist coverage. This result is similar to that obtained
by approximate Bayesian computation. The second contribution compares the
computational efficiency of Bayesian synthetic likelihood and approximate
Bayesian computation using the acceptance probability for rejection and
importance sampling algorithms with a "good" proposal distribution. We show
that Bayesian synthetic likelihood is computationally more efficient than
approximate Bayesian computation, and behaves similarly to regression-adjusted
approximate Bayesian computation. Based on the asymptotic results, the third
contribution proposes using adjusted inference methods when a possibly
misspecified form is assumed for the covariance matrix of the synthetic
likelihood, such as diagonal or a factor model, to speed up the computation.
The methodology is illustrated with some simulated and real examples.
