 Solution

For component $m$, stack its $N_m$ assigned observations as $y_1^{(m)},\dots ,y_{N_m}^{(m)}\in {\mathbb{R}}^n$. The prior is $f_m\sim N_n(0,K)$ and each assigned vector is conditionally $N_n(f_m,{\sigma }^2{I}_n)$. Normal-normal conjugacy giveswhereIf $N_m=0$, this reduces to the prior $N_n(0,K)$.