= Independent normal and inverse-gamma regression priors
With independent <prior distributions> $\beta\sim N(\mu,\Sigma)$ and $s\sim\operatorname{IG}(a,b)$ in a <normal distribution> regression model, the conditional <posterior distribution> of $\beta$ has <covariance matrix> $V=(X^TX/s+\Sigma^{-1})^{-1}$ and <mean> $V(X^Ty/s+\Sigma^{-1}\mu)$. The conditional <posterior distribution> of $s$ is <inverse-gamma distribution> $\operatorname{IG}(a+n/2,b+\|y-X\beta\|^2/2)$. 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>.
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