A Bayesian Interpretation of Pretesting

Summary The usual posterior mean or mode of a k-dimensional regression coefficient is shown to be (1) a weighted average of 2k constrained least-squares points, (2) a weighted average of k + 1 “ideal” points and (3) a weighted average of k + 1 principal component points. These results are used to interpret pretesting procedures which select a constrained least-squares estimate. For special classes of priors a somewhat distorted Bayesian analysis leads to a posterior measure of location that is an F-weighted average of restricted and unrestricted least squares. A spherical prior can be associated with methods that drop principal components. Multilevel testing is shown to imply a non-traditional prior.