Bayesian analysis of finite population sampling in multivariate co-exchangeable structures with separable covariance matric
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We explore the effect of finite population sampling in design problems with many variables cross-classified in many ways. In particular, we investigate designs where we wish to sample individuals belonging to different groups for which the underlying covariance matrices are separable between groups and variables. We exploit the generalised conditional independence structure of the model to show how the analysis of the full model can be reduced to an interpretable series of lower dimensional problems. The types of information we gain by sampling are identified with the orthogonal canonical directions. We first solve a variable problem, which utilises the powerful properties of the adjustment of second-order exchangeable vectors, which has the same qualitative features, represented by the underlying canonical variable directions, irrespective of chosen group, population size or sample size. We then solve a series of group problems which in a balanced design reduce to the sampling of second-order exchangeable vectors. If the population sizes are finite then the qualitative and quantitative features of each group problem will depend upon the sampling fractions in each group, mimicking the infinite problem when the sampling fractions in each group are the same.
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