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  "body": "\n## Definition\n\nLet $K \\ge 1$ be an integer and let $\\set{f_\\theta : \\theta \\in \\Theta}$ be a parametric family of [[probability densities|0ZY9DX8FBF]] on $\\R^p$.\nA *finite mixture model* with $K$ components is the [[statistical model|K3AMMSY8DD]] of the densities\n$$f(x) = \\sum_{k=1}^K \\pi_k f_{\\theta_k}(x), \\qquad x \\in \\R^p,$$\nwhere $\\theta_1, \\ldots, \\theta_K \\in \\Theta$ and the *mixing weights* satisfy $\\pi_k \\ge 0$ and $\\pi_1 + \\cdots + \\pi_K = 1$.\nThe densities $f_{\\theta_k}$ are the *components*.\n\n## Remarks\n\nEquivalently, let $Z$ be a random label with $P(Z = k) = \\pi_k$, and let $X$ have density $f_{\\theta_k}$ given $Z = k$.\nThen $X$ has density $f$, so a mixture model is a [[latent variable model|C63RVH96TF]] with hidden label $Z$.\nExample: $f(x) = \\tfrac{1}{2} \\varphi(x) + \\tfrac{1}{2} \\varphi(x - 3)$, where $\\varphi$ is the standard normal density, is a two-component Gaussian mixture with $\\pi_1 = \\pi_2 = 1/2$.\nNon-example: $\\tfrac{1}{2} \\varphi(x) + \\tfrac{1}{4} \\varphi(x - 3)$ has weights adding up to $3/4$, so it is not a mixture and not even a probability density.\nThe parameters are identifiable at best up to a permutation of the components, since relabelling the components does not change $f$.\nThe same definition with discrete distributions replacing densities gives mixtures of discrete [[distributions|0DVDCMA2DK]].\n",
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