DECOMPOSITION AND BAYESIAN-ANALYSIS OF INVARIANT NORMAL LINEAR-MODELS

Using the theory of linear group representations, we analyse the normal linear model with known sampling covariance structure invariant under a symmetry group, and sampling mean structure equivariant under the same group. In particular, assuming an invariant normal prior distribution on the parameter space, the problem of Bayesian inference is shown to decompose naturally into several independent subproblems. Within any such subproblem, if additional irreducibility conditions hold, it is shown that the posterior expectation of any parameter is a fixed scalar multiple of its unique unbiased estimator, and similarly, the posterior covariance of any two parameters is a fixed scalar multiple of the prior covariance. The theoretical framework is illustrated with reference to experimental designs.