The M step for missing data imputation using MVN updates $$\mu^t=\frac{1}{N_D}\sum_n E[y_n]\\ \Sigma^t=\frac{1}{N_d}\sum_nE[y_n y_n^\top]-\mu^t(\mu^t)^\top $$ where $y_n=(x_n,z_n )$. However, the expectations $E[y_n]$ and $E[y_ny_n^\top]$ depends on $m_n$ and $V_n$ from the following: form

Here, $z_n$ are the hidden variables and $x_n$ are the visible variables. The formula is derived from Gaussian conditionals.

How do I solve for $m_n$ and $V_n$?

Looking closely, they depend on the quantities $\mu_h,\ \ \Sigma_{hh},\ \Sigma_{hv}$ from the Gaussian distribution of the hidden data.

Edit: I think that I can obtain these parameters for the conditional using the same parameter that I obtain from the previous EM iteration ($u^t$ and $\Sigma^t$).