  Approximate Bayesian computation (ABC) refers to a family of inference
methods used in the Bayesian analysis of complex models where evaluation of the
likelihood is difficult. Conventional ABC methods often suffer from the curse
of dimensionality, and a marginal adjustment strategy was recently introduced
in the literature to improve the performance of ABC algorithms in
high-dimensional problems. The marginal adjustment approach is extended using a
Gaussian copula approximation. The method first estimates the bivariate
posterior for each pair of parameters separately using a 2-dimensional Gaussian
copula, and then combines these estimates together to estimate the joint
posterior. The approximation works well in large sample settings when the
posterior is approximately normal, but also works well in many cases which are
far from that situation due to the nonparametric estimation of the marginal
posterior distributions. If each bivariate posterior distribution can be well
estimated with a low-dimensional ABC analysis then this Gaussian copula method
can extend ABC methods to problems of high dimension. The method also results
in an analytic expression for the approximate posterior which is useful for
many purposes such as approximation of the likelihood itself. This method is
illustrated with several examples.
