Proposition 8.1. For the linear model (8.5) with conjugate prior (8.6), the posterior distribution for (B, a) after observing responses y = (V1, ..., yn) corresponds to Blo,X,y ~ Normal(mn, J'V.), o' | X,y ~ Gamma(an, br), where V =(V-] + X X) ', mn = Vn XTy, n an =a+ br =b+ =(y y -yXmm). (8.10)\fThe linear model (1) can be equivalently expressed in terms of independent and identically distributed error variables e = (61, ..., (.), y = XP+E, 6 - Normal(0, o'), i=1,..., n. i. Still assuming the conjugate prior Bayes linear model, and starting from the posterior distribution from Proposition 8.1 for the parameters (8,