When performing a Bayesian analysis on the Olympics data, we assumed that the prior was known. If a Gaussian prior is placed on \(\mathbf{w}\) and an inverse gamma prior on the variance \(\sigma^{2}\)

\[p\left(\sigma^{2} \mid \alpha, \beta\right)=\frac{\beta^{\alpha}}{\Gamma(\alpha)}\left(\sigma^{2}\right)^{-\alpha-1} \exp \left\{-\frac{\beta}{\sigma^{2}}\right\}\]

the posterior will also be the product of a Gaussian and an inverse gamma. Compute the posterior parameters.