The local independence postulate plays a central role in latent variable models. Local independence means that all the manifest variables are independent random variables if the latent variables are controlled (fixed).
Technically, the local independence may be described by formula
| P(y1, … ,yL | x) = | L Õ l=1 | Pl(yl | x) |
where (y1, … ,yL) is the vector of all the manifest variables, x is the latent variable, P(·|x) is the conditional probability for y=(y1,…,yL) given x ; Pl(·|x) are conditional probabilities for each manifest variable yl separately. If the manifest variables {yl} are continuous, then P(·|x) and Pl( ·| X ) are probability densities, not probabilities.