Based on the model that is specified in the section Notation for the Item Response Theory Model, the marginal likelihood is
where , is the multivariate normal density function for the latent factor , and is a set of all the model parameters. The corresponding log likelihood is
Integrations in the preceding equation cannot be solved analytically and need to be approximated by using numerical integration,
where d is the number of factors, G is the number of quadrature points per dimension, and and are the quadrature points and weights, respectively.