Gaussian mixture model
/ GOW-see-an MIKS-chur MOD-ul /
A Gaussian mixture model assumes your data was produced by blending several simple bell-shaped sources, and tries to reverse-engineer them. Picture the heights of everyone in a town: the single histogram has two humps, because it secretly mixes a shorter bell curve (children) and a taller one (adults). A GMM looks at the combined pile and figures out: how many bell curves are hiding in here, where is each one centered, how wide is it, and what fraction of the data came from each? It's clustering, but built on probability.
A bell curve — the Gaussian, or normal distribution — is the familiar hump where most values sit near an average and grow rarer toward the edges. A mixture model says the data is a weighted sum of several such bells. Crucially, instead of forcing each point firmly into one group the way k-means does, a GMM gives soft memberships: it might say a given person is 70% likely from the "adult" bell and 30% from the "child" bell. It learns all this by an elegant back-and-forth (the EM algorithm): guess the bells, assign points softly, re-fit the bells to those assignments, repeat until stable.
Its advantage over k-means is flexibility: the bells can be stretched and tilted into ellipses, so it captures oval and overlapping clusters that k-means' rigid circles would butcher, and the soft memberships honestly express uncertainty at the borders. The catch is that you still must choose how many bells in advance, it assumes the underlying groups really are bell-shaped (a real assumption that can be wrong), and like k-means it can settle into a poor answer depending on where it started, so multiple runs are wise.
A histogram of customer spending has two bumps. A GMM untangles them into two bell curves: a "casual" group centered at $30 (wide) and a "premium" group centered at $120 (narrow), with 80% of customers from the first. A shopper who spent $75 is reported as 55% casual, 45% premium — honest about the in-between.
Soft memberships: a point can belong partly to several bells at once.
Think of it as k-means' more flexible, probabilistic cousin: it allows tilted oval clusters and gives soft "how likely" memberships instead of hard assignments. But it leans on the assumption that the groups are genuinely bell-shaped — when they aren't, its tidy ellipses can mislead.