This note derives the posterior, evidence, and predictive density for linear multivariate regression under zero-mean Gaussian noise. Many Bayesian texts, such as Box & Tiao (1973), cover linear regression. This note contributes to the discussion by paying careful attention to invariance issues, demonstrating model selection based on the evidence, and illustrating the shape of the predictive density. Piecewise regression and basis function regression are also discussed.
This note assumes you have read "Inferring a Gaussian distribution."