Consider vectors of item responses obtained from a sample of subjects from a population in which ability θ is distributed with density g (θ‖α), where the α are unknown parameters. Assuming the responses depend on θ through a fully specified item response model, this paper presents maximum likelihood equations for the estimation of the population parameters directly from the observed responses; i.e., without estimating an ability parameter for each subject. Also provided are asymptotic standard errors and tests of fit, computing approximations, and details of four special cases: a non-parametric approximation, a normal solution, a resolution of normal components, and a beta-binomial solution.