Structural Probability and Prediction for the Multivariate Model

Summary The multivariate normal distribution can be used to describe the response variable of a system. A more comprehensive multivariate model is described in this paper: it has a distribution describing an error variable internal to a system, with a known multivariate distribution; and it has a positive affine transformation, the physical quantity, which generates a response vector from an error vector. This more comprehensive model is a structural model and it provides structural probability statements concerning the physical quantity. Error and structural distributions are derived for the multivariate model. The structural distribution for a quantity can be used to generate structural prediction distributions : various prediction distributions are obtained for the multivariate model. The results are specialized to cover the case of the multivariate normal structural model. The classical multivariate normal has been analysed by Bayesian methods (Geisser and Cornfield, 1963). The more comprehensive multivariate structural model does not need the use of subjective methods.