 Independent normal and inverse-gamma regression priors

With independent prior distributions $\beta \sim N(\mu ,\Sigma )$ and $s\sim \mathrm{IG}(a,b)$ in a normal distribution regression model, the conditional posterior distribution of $\beta$ has covariance matrix $V=(X^{T}X/s+{\Sigma }^{-1}{)}^{-1}$ and mean $V(X^{T}y/s+{\Sigma }^{-1}\mu )$. The conditional posterior distribution of $s$ is inverse-gamma distribution $\mathrm{IG}(a+n/2,b+\Vert y-X\beta {\Vert }^2/2)$. The shape contains no half-dimension contribution from $\beta$, since its prior distribution is independent of $s$. These conditionals give a two-block Gibbs sampler.