Derive the MAP update for a mixture model with Gaussian components that are independent over the \(D\) dimensions

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

assuming an independent Gaussian prior on each \(\mu_{k d}\) with mean \(m\) and variance \(s^{2}\).