  Bayes linear analysis and approximate Bayesian computation (ABC) are
techniques commonly used in the Bayesian analysis of complex models. In this
article we connect these ideas by demonstrating that regression-adjustment ABC
algorithms produce samples for which first and second order moment summaries
approximate adjusted expectation and variance for a Bayes linear analysis. This
gives regression-adjustment methods a useful interpretation and role in
exploratory analysis in high-dimensional problems. As a result, we propose a
new method for combining high-dimensional, regression-adjustment ABC with
lower-dimensional approaches (such as using MCMC for ABC). This method first
obtains a rough estimate of the joint posterior via regression-adjustment ABC,
and then estimates each univariate marginal posterior distribution separately
in a lower-dimensional analysis. The marginal distributions of the initial
estimate are then modified to equal the separately estimated marginals, thereby
providing an improved estimate of the joint posterior. We illustrate this
method with several examples. Supplementary materials for this article are
available online.
