Bayes linear covariance matrix adjustment

Details Bayes linear covariance matrix adjustment Bayes linear covariance matrix adjustment

1995-12-04

arxiv.org/abs/bayes-an/9512001v1

Physics

Data Analysis, Statistics and Probability

146 pages. PhD thesis. Available as Postscript only. Also available from http://fourier.dur.ac.uk:8000/~dma1djw/pub/thesis.htm

Scientific paper

In this thesis, a Bayes linear methodology for the adjustment of covariance matrices is presented and discussed. A geometric framework for quantifying uncertainties about covariance matrices is set up, and an inner-product for spaces of random matrices is motivated and constructed. The inner-product on this space captures aspects of our beliefs about the relationship between covariance matrices of interest to us, providing a structure rich enough for us to adjust beliefs about unknown matrices in the light of data such as sample covariance matrices, exploiting second-order exchangeability and related specifications to obtain representations allowing analysis. Adjustment is associated with orthogonal projection, and illustrated with examples of adjustments for some common problems. The problem of adjusting the covariance matrices underlying exchangeable random vectors is tackled and discussed. Learning about the covariance matrices associated with multivariate time series dynamic linear models is shown to be amenable to a similar approach. Diagnostics for matrix adjustments are also discussed.

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