Exercise 12.10 - Efficiently evaluating the PPCA density

We firstly complete the proof of Theorem 12.2.2, this can be done by plugging (12.61) back into (12.60). Recall that the MLE is where $\mathbf{\text{𝐖}}$ satisfies (using $\frac{\partial {\mathbf{\text{𝐀}}}^{-1}}{\partial \mathbf{\text{𝐀}}}=-{\mathbf{\text{𝐀}}}^{-1}{\mathbf{\text{𝐀}}}^{-1}$):

With:

we have:

In which $\mathbf{\text{𝐐}}$ is the sigular vectors/eigen vectors for $\mathbf{\text{𝐒}}$. This would reduce $(\mathbf{\text{𝐈}}-\mathbf{\text{𝐒}}{\mathbf{\text{𝐂}}}^{-1}){\mathbf{\text{𝐂}}}^{-1}\mathbf{\text{𝐖}}$ to zero, hence complete the proof. For MLE of the ${\sigma }^2$, using the trace trick.

Now for $p(\mathbf{\text{𝐱}}|\widetilde{\mathbf{\text{𝐖}}},{\tilde{\sigma }}^2)$, it is a normal distribution with zero mean and covariance whose last $L-D$ components are uniformly ${\tilde{\sigma }}^2$.