  We describe a numerical scheme for evaluating the posterior moments of
Bayesian linear regression models with partial pooling of the coefficients. The
principal analytical tool of the evaluation is a change of basis from
coefficient space to the space of singular vectors of the matrix of predictors.
After this change of basis and an analytical integration, we reduce the problem
of finding moments of a density over k + m dimensions, to finding moments of an
m-dimensional density, where k is the number of coefficients and k + m is the
dimension of the posterior. Moments can then be computed using, for example,
MCMC, the trapezoid rule, or adaptive Gaussian quadrature. An evaluation of the
SVD of the matrix of predictors is the dominant computational cost and is
performed once during the precomputation stage. We demonstrate numerical
results of the algorithm. The scheme described in this paper generalizes
naturally to multilevel and multi-group hierarchical regression models where
normal-normal parameters appear.
