A worked numerical example tracks one full Expectation-Maximization cycle on two points, x=0 and x=2, under a two-component Gaussian mixture. The E-step freezes parameters and computes responsibilities showing how strongly each point belongs to each cluster. The M-step then uses those weighted responsibilities to update means, variances, and mixing weights simultaneously. Animated connections visualize the weighting, while old and new parameter values are clearly contrasted. Useful for first-year AI students learning why EM monotonically improves likelihood without guaranteeing global optimality.
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# EM alternates between responsibilities and parameter estimates Use a deliberately small Gaussian-mixture calculation with samples x = [0,2], initial means 0 and 2, equal mixing weights, and variances 1. During the E-step, freeze the parameters and compute responsibility rows [0.880797,0.119203] and [0.119203,0.880797]. During the M-step, use these weights to update the means to approximately 0.238406 and 1.761594; both variances become approximately 0.419974 and the mixing weights remain one half. Animate responsibility-weighted connections to the centers, then update all parameters together. Link the operations to the observed-data log likelihood and explain why exact EM under its standard conditions does not decrease that objective, without guaranteeing a global optimum. Identify this as a one-iteration arithmetic example, not a dataset for demonstrating stable long-term fitting; covariance collapse can occur. Use black backgrounds and carefully distinguish old and new parameter values. Audience: 一年级人工智能专业学生