Latent variable model

Remarks

Often the components $X_1, \ldots, X_m$ of $X$ are assumed conditionally independent given $Z$, the assumption of local independence. Example: let $Z$ be standard normal and, given $Z$, let $X_1, X_2$ be independent with distribution $N(Z, 1)$. Then $(X_1, X_2)$ is bivariate normal with variances $2$ and covariance $1$; the dependence of $X_1$ and $X_2$ comes only from $Z$. Non-example: a regression model of $Y$ given a measured covariate $x$ is not a latent variable model, since all of its variables are observed. The models are classified by the type of the variables. Factor analysis has continuous latent and observed variables. Latent class and mixture models have discrete latent variables. Item response models have continuous latent and discrete observed variables.