Approximate Bayesian computation and Bayes linear analysis: Towards highdimensional ABC
Abstract
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 regressionadjustment ABC algorithms produce samples for which first and second order moment summaries approximate adjusted expectation and variance for a Bayes linear analysis. This gives regressionadjustment methods a useful interpretation and role in exploratory analysis in highdimensional problems. As a result, we propose a new method for combining highdimensional, regressionadjustment ABC with lowerdimensional approaches (such as using MCMC for ABC). This method first obtains a rough estimate of the joint posterior via regressionadjustment ABC, and then estimates each univariate marginal posterior distribution separately in a lowerdimensional 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.
 Publication:

arXiv eprints
 Pub Date:
 December 2011
 arXiv:
 arXiv:1112.4755
 Bibcode:
 2011arXiv1112.4755N
 Keywords:

 Statistics  Methodology;
 Statistics  Computation
 EPrint:
 To appear in Journal of Computational and Graphical Statistics