 Solution

Let $v={\sigma }^2>0$, $d_k=k+1$, and $S_k(a)=\Vert y-X_{k}a{\Vert }^2$. Under the standard interpretation that the two stated priors are independent, multiply the normal likelihood by the normal coefficient prior and the inverse-gamma distribution density. The independent normal and inverse-gamma regression priors giveFor a fixed model the determinant and Gaussian normalizing constants are independent of $a_k,v$ and may be omitted here; ${\Sigma }_k$ is assumed positive-definite. The coefficient prior is not scaled by $v$, so replacing its precision by ${\Sigma }_k^{-1}/v$ would describe a different prior.

For an additional check, completing the coefficient square and reading the variance kernel give the full conditional distributionsandMarginal priors alone would not determine their joint prior without the independence assumption. The front-page inverse-gamma mean has a typographical error: this density has mean $\beta /(\alpha -1)$ when $\alpha >1$, not a denominator $\alpha +1$. The density itself, used in the posterior calculation, fixes the correct parameterization.