The paper is devoted to discussing consequences of the so-called Frisch-Waugh Theorem to posterior inference and Bayesian model comparison. We adopt a generalised normal linear regression framework and weakenits assumptions in order to cover non-normal, jointly elliptical samplingdistributions, autoregressive specifications, additional nuisance parameters andmulti-equation SURE or VAR models. The main result is that inference basedon the original full Bayesian model can be obtained using transformed dataand reduced parameter spaces, provided the prior density for scale or precisionparameters is appropriately modified.