Question: 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

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

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

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