The IRT Procedure
Based on the model that is specified in the section Notation for the Item Response Theory Model, the marginal likelihood is
![\[ L(\btheta |U) = \prod _{i=1}^ N \int \prod _{j=1}^ J \prod _{k=1}^ K(P_{ijk})^{v_{ijk}} \phi (\bm {\eta };\bmu ,\bSigma )d\bm {\eta } = \prod _{i=1}^ N \int f(u_ i|\bm {\eta }) \phi (\bm {\eta };\bmu ,\bSigma )d\bm {\eta } \]](images/statug_irt0058.png){kind=small}
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
![\[ \log L(\btheta |U) = \sum _{i=1}^ N \log \int \prod _{j=1}^ J \prod _{k=1}^ K(P_{ijk})^{v_{ijk}} \phi (\bm {\eta };\bmu ,\bSigma )d\bm {\eta } \]](images/statug_irt0063.png){kind=small}
Integrations in the preceding equation cannot be solved analytically and need to be approximated by using numerical integration,
![\[ \log \tilde{L}(\btheta |U) = \sum _{i=1}^ N \log \left[ \sum _{g=1}^{G^ d}\left[\prod _{j=1}^ J \prod _{k=1}^ K(P_{ijk}(\mb{x}_ g))^{v_{ijk}} \frac{\phi (\mb{x}_ g;\bmu ,\bSigma )}{\phi (\mb{x}_ g;0,I)}\right]w_ g\right] \]](images/statug_irt0064.png){kind=small}
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.