Overview
A latent variable or factor model is a model where the variables of interest, here labeled {% x_1,x_2, ... ,x_n %} are hypothesized to have a relationship to another set of variables, labeled {% y_1,y_2,...,y_m %}. These variable, called the latent variables, may or may not be observable. That is, it is possible to assume a model where the latent variables are unobserved, but still provide structure to the model.
The variables of interest are assumed to have distribution conditional on the latent variables.
{% x_i \sim f_i(x_i|\vec{y}) %}
The latent variables will typically be assumed to follow some distribution (sometimes called the prior distribution)
{% \vec{y} \sim g(\vec{y}) %}
Reasons for Using Latent Variables
- Latent variables can provide explanations for why the observed variables are measured at their current values
- Latent variables can simplify calculations. In particular, variables that are correlated can be more easily analyzed if the correlation is structured to come from a latent variable.