 EM maximization step

Maximize $Q(\theta \mid {\theta }^{(t)})$ over the permitted statistical parameters, with the E-step conditional distribution held fixed.^[ref] A finite mixture model hasFor normal distributions with known common variance ${\sigma }^2>0$ and fixed positive mixture weights, only the means are fitted. Dropping terms independent of these candidate means givesThe derivative in ${\mu }_j$ is ${\sum }_i{r}_{ij}^{(t)}(y_i-{\mu }_j)/{\sigma }^2$, and its second derivative is $-N_j^{(t)}/{\sigma }^2$. Thus a positive weighted count gives a unique maximizing weighted mean. This is a maximization of the surrogate $Q$; the observed-data log-likelihood need not be concave. Recompute the mixture responsibilities after updating the means. The broader EM for Gaussian mixtures with a common variance also estimates mixture weights and the shared variance.