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  "body": "\n## Definition\n\nLet $\\Theta$ be a parameter set.\nLet $(X, Z)$ be a pair of random vectors, where $X$ is observed and $Z$ is not.\nA *latent variable model* is a family $\\set{P_\\theta : \\theta \\in \\Theta}$ of [[joint distributions|0DR81R1F2B]] of $(X, Z)$.\nThe variables $Z$ are the *latent variables*.\nThe [[statistical model|K3AMMSY8DD]] for the data is the family of [[marginal distributions|0MXT6PRZZ6]] of $X$ under $P_\\theta$.\nWhen $P_\\theta$ has a density $p_\\theta(x, z)$ with respect to a product measure $\\mu \\otimes \\nu$, the observed density is\n$$p_\\theta(x) = \\int p_\\theta(x, z) \\, d\\nu(z).$$\n\n## Remarks\n\nOften the components $X_1, \\ldots, X_m$ of $X$ are assumed [[conditionally independent|0THJ7416DX]] given $Z$, the assumption of local independence.\nExample: let $Z$ be standard normal and, given $Z$, let $X_1, X_2$ be independent with distribution $N(Z, 1)$.\nThen $(X_1, X_2)$ is bivariate normal with variances $2$ and covariance $1$; the dependence of $X_1$ and $X_2$ comes only from $Z$.\nNon-example: a regression model of $Y$ given a measured covariate $x$ is not a latent variable model, since all of its variables are observed.\nThe models are classified by the type of the variables.\nFactor analysis has continuous latent and observed variables.\nLatent class and mixture models have discrete latent variables.\nItem response models have continuous latent and discrete observed variables.\n",
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