3.12 ( ) We saw in Section 2.3.6 that the conjugate prior for a Gaussian distribution with unknown mean and unknown precision (inverse variance) is a normal-gamma distribution. This property also holds for the case of the conditional Gaussian distribution p(t|x,w, β) of the linear regression model. If we consider the likelihood function (3.10), then the conjugate prior for w and β is given by p(w, β) = N(w|m0, β

−1S0)Gam(β|a0, b0). (3.112)

Show that the corresponding posterior distribution takes the same functional form, so that p(w, β|t) = N(w|mN, β
−1SN)Gam(β|aN, bN) (3.113)
and find expressions for the posterior parameters mN, SN, aN, and bN.