= EM for Gaussian mixtures with a common variance
{c}
{title2=$(\pi_j^{\mathrm{new}},\mu_j^{\mathrm{new}},\sigma^{2\mathrm{new}})$}
The E-step computes <mixture responsibilities> $\tau_{ij}$ from the old parameters. Put $N_j=\sum_i\tau_{ij}$. The M-step updates $\pi_j=N_j/n$, $\mu_j=\sum_i\tau_{ij}y_i/N_j$ and $\sigma^2=\sum_{i,j}\tau_{ij}(y_i-\mu_j^{\mathrm{new}})^2/n$. The variance uses new means and old responsibilities. <EM likelihood monotonicity> guarantees nondecrease of the observed likelihood, not a global optimum.
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