Derive the EM update for the variance of the \(d\) th dimension and the \(k\) th component, \(\sigma_{k d}^{2}\), when the cluster components have a diagonal Gaussian likelihood

\[p\left(\mathbf{x}_{n} \mid z_{n k}=1, \mu_{k 1}, \ldots, \mu_{K D}, \sigma_{k 1}^{2}, \ldots, \sigma_{k D}^{2}\right)=\prod_{d=1}^{D} \mathcal{N}\left(\mu_{k d}, \sigma_{k d}^{2}\right)\]