Two or three overlapping bell curves summed into a multi-modal distribution.
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Mixture of Gaussians
A Gaussian mixture model (GMM) represents a population as the weighted sum of two or more normal distributions. The faint dashed curves are the individual components, each with its own mean, standard deviation, and mixing weight. The bold solid curve is their weighted sum: the distribution you would observe sampling from the population without knowing which component generated each point.\n\nThe result is multi-modal. Two humps appear when the component means are far enough apart relative to their standard deviations. The valley between the humps forms a natural decision boundary for separating the two subgroups. When components overlap heavily, the mixture looks like a single broadened distribution, which is why naively assuming normality can conceal latent subgroups.\n\nExample: heights of male and female adults in a combined sample form a bimodal mixture. Neither subpopulation is non-normal; it is the mixing that produces two peaks.
Fitting a GMM to data requires estimating each component mean, standard deviation, and mixing weight from observations alone. The standard method is the Expectation-Maximization (EM) algorithm. In the E-step, each data point receives a soft probability of belonging to each component given current parameter estimates. In the M-step, those soft assignments update all parameters to maximize the likelihood. The two steps alternate until convergence.\n\nDempster, Laird, and Rubin formalized EM in 1977. GMMs built on this framework appear in clustering, speaker identification, and density estimation pipelines across statistics and machine learning.
Maximum Likelihood from Incomplete Data via the EM Algorithm , Dempster, A. P., Laird, N. M., and Rubin, D. B. (1977) - Journal of the Royal Statistical Society, Series B, 39(1), 1-38
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This mixture of gaussians is an editorial illustration built to represent the concept accurately. Where it shows figures, they are typical or representative values chosen to make the relationship clear, not a single underlying dataset. The diagram and its explainer are reviewed and maintained centrally, and updated over time as understanding improves.
Superimposing multiple bell curves
A mixture of Gaussians combines two or more normal distributions, each with its own mean and standard deviation, weighted by mixture proportions that sum to 1. The result is a multi-modal probability distribution where the overall shape reflects contributions from all component distributions.
If two populations are sampled together without labels, the combined data might exhibit two distinct peaks. A mixture model captures this structure: perhaps 40 percent of observations come from group A (mean = 20, SD = 3) and 60 percent from group B (mean = 35, SD = 4).
Identifying latent groups and model selection
Mixture models are useful for clustering and segmentation when you suspect hidden subgroups but lack explicit labels. Machine learning algorithms like expectation-maximization (EM) estimate both the means and proportions of each component from data alone.
A challenge is choosing the right number of components. Too few components under-represent the data's complexity; too many introduce spurious detail. Information criteria (AIC, BIC) or cross-validation help balance fit against model complexity. Visualizing the overlaid component curves often reveals whether the model makes intuitive sense.
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