Approximate Bayesian computation and Bayes linear analysis: Towards high-dimensional ABC

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.