Exercise 7.10 - Bayesian linear regression using the g-prior

For Bayesian linear regression model, the likelihood is as always:

The prior distribution is Gaussian-Inverse Gamma distribution:

The posterior distribution takes the form:

To decompose the updated hyperparameters from this form, we have to find:

• The exponential of ${\sigma }^2$.
• The squared term within the exponential.

The exponential of ${\sigma }^2$ in the posterior is:

thus we have:

The coefficient of ${\mathbf{\text{𝐰}}}^{\text{T}}\mathbf{\text{𝐰}}$ in the exponential (as the introducer matrix of the inner product) is:

therfore:

with ${\mathbf{\text{𝐕}}}_0=g{(\mathbf{\text{𝐗}}{\mathbf{\text{𝐗}}}^{\text{T}})}^{-1}$:

The coefficient of ${\mathbf{\text{𝐰}}}^{\text{T}}$ in the exponential is:

So we have:

This yields:

Finally, completing the square within the exponential yields:

Plugging in what we have already known about ${\mathbf{\text{𝐰}}}_N$ and ${\mathbf{\text{𝐕}}}_N$ results in:

Now we have (7.113)-(7.116) established.