A two-component Gaussian mixture (means 0 and 2, variance 1, equal weights) is built up to show how combining simple bell curves explains a dataset with more than one peak. The animation contrasts prior mixing weights with posterior responsibilities computed at specific points, including an exact calculation at x=0. Three side panels compare K-means hard assignment, fuzzy membership, and Gaussian-mixture responsibility, clarifying soft versus hard clustering for students starting in AI and machine learning.
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# A mixture of Gaussians explains more than one peak Start with a dataset whose shape is not well described by a single Gaussian. Build a mixture from two one-dimensional Gaussian components with equal mixing weights, means 0 and 2, and variances 1. Draw each weighted density and their sum. Explain the generative process: first select a component, then draw a sample from that component. At x = 1, the two posterior responsibilities are equal. At x = 0, calculate responsibilities approximately 0.880797 and 0.119203 by normalizing the weighted densities at that location. Keep prior mixing weights distinct from these sample-specific responsibilities. Compare a K-means hard assignment, a fuzzy membership, and a Gaussian-mixture responsibility using three small panels. Component identifiers should remain abstract rather than becoming assumed semantic classes. Use black backgrounds, cyan and orange curves, and density-labeled axes. Useful for understanding latent components and probabilistic soft assignment. Audience: 一年级人工智能专业学生