GMM responsibilities and M-step

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Problem

Two GMM components have weights $(0.25,0.75)$ and densities at $x$ of $(0.6,0.2)$ . Compute the responsibilities. Then, observations $(0,10)$ have responsibility rows $(0.5,0.5)$ and $(0.8,0.2)$ . Compute the updated weights and means.

Reveal answer or reference solution

The unnormalized values are $(0.15,0.15)$ , so the responsibilities are $(0.5,0.5)$ . The effective counts over the two observations are $(1.3,0.7)$ , giving weights $(0.65,0.35)$ . The means are

\frac{0.5(0)+0.8(10)}{1.3}=\frac{80}{13}\qquad\text{and}\qquad \frac{0.5(0)+0.2(10)}{0.7}=\frac{20}{7}.

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GMM responsibilities and M-step

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