  Likelihood-free methods are an established approach for performing
approximate Bayesian inference for models with intractable likelihood
functions. However, they can be computationally demanding. Bayesian synthetic
likelihood (BSL) is a popular such method that approximates the likelihood
function of the summary statistic with a known, tractable distribution --
typically Gaussian -- and then performs statistical inference using standard
likelihood-based techniques. However, as the number of summary statistics
grows, the number of model simulations required to accurately estimate the
covariance matrix for this likelihood rapidly increases. This poses significant
challenge for the application of BSL, especially in cases where model
simulation is expensive. In this article we propose whitening BSL (wBSL) -- an
efficient BSL method that uses approximate whitening transformations to
decorrelate the summary statistics at each algorithm iteration. We show
empirically that this can reduce the number of model simulations required to
implement BSL by more than an order of magnitude, without much loss of
accuracy. We explore a range of whitening procedures and demonstrate the
performance of wBSL on a range of simulated and real modelling scenarios from
ecology and biology.
