# The IRT Procedure

### Approximating the Marginal Likelihood

Subsections:

As discussed in the section Marginal Likelihood, integrations that are involved in the marginal likelihood for IRT model cannot be solved analytically and need to be approximated by using numerical integration, mostly Gauss-Hermite quadrature.

In general, the Gauss-Hermite (G-H) quadrature can be presented as

where G is the number of quadrature points and and are the integration points and weights, respectively, which are uniquely determined by the integration domain and the weighting kernel . Traditional G-H quadrature often uses as the weighting kernel. In the field of statistics, the density of standard normal distribution is more widely used instead, because for estimating various statistical models, the Gaussian density is often a factor of the integrand. In the case in which the Gaussian density is not a factor of the integrand, the integral is transformed into the form by dividing and multiplying the original integrand by the standard normal density.