Simplifying Bayesian experimental design for multivariate partially exchangeable systems
We adopt a Bayes linear approach to tackle design problems with many variables cross-classified in many ways. We investigate designs where we wish to sample individuals belonging to different groups, exploiting the powerful properties of the adjustment of infinitely second-order exchangeable vectors. The types of information we gain by sampling are identified with the orthogonal canonical directions. We show how we may express these directions in terms of the different factors of the model. This allows us to solve a series of lower dimensional problems, through which we may identify the different aspects of our adjusted beliefs with the different aspects of the choice of design, leading both to qualitative insights and quantitative guidance for the optimal choice of design. These subproblems have an interpretable form in terms of adjustment upon subspaces of the full problem and remain valid when we consider adjusting the underlying population structure and also for predicting future observables from past observation. We then examine the adjustment of finitely second-order exchangeable vectors, and show that the adjustment shares the same powerful properties as the adjustment in the infinite case. We show how if the finite sequence of vectors is extendible, then the differences in the adjustment of the sequence is quantitatively the same for all sequence lengths and it is easy to compare the qualitative differences. Extending to an infinite sequence allows us to draw comparisons between the finite and infinite modelling. Such comparisons may also be made when we consider sampling individuals belonging to different groups, where each group contains only a finite number of individuals.