Exercise 24.5 - Gibbs sampling for robust linear regression with a Student t likelihood

Recall that in robust linear regression, each sample $y_i$ is assigned an extra latente variable $z_i$ as the estimation of the noise. The complete likelihood is:

From which we can read that the conditional distribution for $z_n$ takes the form:

hence is again a Gamma distribution.

For the variance ${\sigma }^2$, its conditional posterior is:

which can be absorbed into the prior with an Inverse-Gamma form.

Finally, for the weight $\mathbf{\text{𝐰}}$, the conditional posterior takes the Gaussian form, so using a Gaussian conjugate prior is sufficient for Gibbs sampling update.