Gaussian Mixture Models: The EM Algorithm in Practice

You have a dataset that clearly contains distinct groups, but nobody has labelled them. K-Means would give you hard boundaries — each point belongs to exactly one cluster. But what if a data point sits right between two clusters? Shouldn't we express that uncertainty? Gaussian Mixture Models (GMMs) do exactly this. They model your data as a mixture of Gaussian distributions, assigning each point a probability of belonging to each cluster — not a hard label. By the end of this post, you'll implement a GMM from scratch using the EM algorithm and cluster real data from the Old Faithful geyser. If you haven't read the EM algorithm tutorial yet, don't worry — we'll explain everything you need here. But if you want the gentle coin-toss introduction first, start there. Quick Win: Cluster Old Faithful Eruptions Old Faithful is a geyser in Yellowstone National Park. Its eruptions come in two types: short (~2 min) with short waits, and long (~4.5 min) with long waits. Let's discover these cluste