Exercise 11.12 - EM for robust linear regression with a Student t likelihood

Assuming that $v$ and ${\sigma }^2$ are fixed, then the likelihood takes the form:

We only derive the M-step for this exercise with fixed ${\sigma }^2$ and $v$. The negative logarithm likelihood is:

Taking gradient:

The dependency of the denominator on $\mathbf{\text{𝐰}}$ makes the optimum analytically intractable. One possible solution is to randomly initialize $\mathbf{\text{𝐰}}$ with ${\mathbf{\text{𝐰}}}_0$ and conduct iteration: