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A type of mixed distribution or model which assumes subpopulations follow Gaussian distributions.

Gaussian mixture can refer to distributions or models which assumes subpopulations follow Gaussian distributions. A mixed Gaussian distribution $\mathcal{P}(x)$ can be generally written as a weighted sum of individual Gaussians $\mathcal{N}$:

$$\mathcal{P}(x) = \sum_{i=1}^n w_i \mathcal{N}(\mu_i,\sigma_i^2)$$

where $\sum_{i=1}^{n}w_i = 1$

GMMs are often used in unsupervised learning where we don't know from which subpopulation a data point belongs. In this case we seek to maximize the likelihood function (which assumes the data is independent):

$$p(X|w,\mu,\sigma) = \prod_{i=1}^{n} p(x_i|w,\mu,\sigma)$$

The parameters $\{w,\mu,\sigma^2\}$ can be solved using either Expectation-Maximization (EM) or MAP estimation.

It can also be used to simulate outliers and thereby test outlier detection and robust statistical methods

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