Matrix Weighted Averages and Posterior Bounds

SUMMARY Bounds for matrix weighted averages of pairs of vectors are presented. The weight matrices are constrained to certain classes suggested by the Bayesian analysis of the linear regression model and the multivariate normal model. The bounds identify the region within which the posterior location vector must lie if the prior comes from a certain class of priors. WE present in this article a series of results on the behaviour of matrix weighted averages of the form b** = (H + H*)-1 (Hb + H* b*), where b, b* and b** are vectors and where H and H* are square, symmetric, positive semi- definite matrices. These results indicate the extent to which we can generalize to higher dimensions the trivial univariate bound that constrains the scalar b** to lie algebraically between the scalars b and b*. Our interest in the behaviour of matrix weighted averages derives from the fact that two statistical models-the normal linear regression model and the multivariate normal sampling model with normal priors-have posterior means of the location parameters that are matrix weighted averages of a prior location vector and a sample location vector. One or both of the matrices in these averages are arbitrary either because prior distributions are impossible to measure without error or because intended readers may differ in their prior judgments. A Bayesian analysis based on any particular prior distribution will as a result be of little interest. Practical users of the Bayesian tools will necessarily face the difficult reporting problem of characterizing economically the mapping implied by the given data from interesting prior distributions into their respective posterior distributions, thereby servicing a wide readership as well as identifying those features of the prior which critically determine the posterior and which must therefore be measured accurately. One way of characterizing the mapping from priors into posteriors is a local sensitivity analysis that identifies the relative sensitivity of aspects of the posterior distribution to infinitesimal changes in the prior. The feasibility of a local sensitivity analysis is somewhat doubtful since to have great content it will have to be performed for many different prior distributions. Instead, we are suggesting here a global sensitivity analysis that constructs a correspondence between classes of priors and classes of posteriors. We will attempt to answer questions of the form: "If my prior is a member of this class of priors, what can I say about my posterior?" Although we would naturally be interested in both the location and the dispersion of the posterior, we will consider here only the location parameter. We will take the location of the prior as given and will develop a correspondence between classes of prior covariance matrices and regions in the space of the posterior location vector. We will see that a great deal can be said about the posterior location without precisely specifying the prior covariance matrix.