Bayes Linear Sufficiency in Non-exchangeable Multivariate Multiple Regressions

Abstract

We consider sufficiency for Bayes linear revision for multivariate multiple regression problems, and in particular where we have a sequence of multivariate observations at different matrix design points, but with common parameter vector. Such sequences are not usually exchangeable. However, we show that there is a sequence of transformed observations which is exchangeable and we demonstrate that their mean is sufficient both for Bayes linear revision of the parameter vector and for prediction of future observations. We link these ideas to making revisions of belief over replicated structure such as graphical templates of model relationships. We show that the sufficiencies lead to natural residual collections and thence to sequential diagnostic assessments. We show how each finite regression problem corresponds to a parallel implied infinite exchangeable sequence which may be exploited to solve the sample-size design problem. Bayes linear methods are based on limited specifications of belief, usually means, variances, and covariances. As such, the methodology is well suited to high-dimensional regression problems where a full Bayesian analysis is difficult or impossible, but where a linear Bayes approach offers a pragmatic way to combine judgements with data in order to produce posterior summaries.