Repeat Exercise 6.1 with isotropic Gaussian components: [pleft(mathbf{x}_{n} mid z_{n k}=1, boldsymbol{mu}_{k}, sigma_{k}^{2} ight)=prod_{d=1}^{D} mathcal{N}left(mu_{k d}, sigma_{k}^{2}

Repeat Exercise 6.1 with isotropic Gaussian components:

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

Data from Exercise 6.1

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

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