Automatic approximation of the marginal likelihood in non-Gaussian hierarchical models

Fitting of non-Gaussian hierarchical random effects models by approximate maximum likelihood can be made automatic to the same extent that Bayesian model fitting can be automated by the program BUGS. The word “automatic” means that the technical details of computation are made transparent to the user. This is achieved by combining a technique from computer science known as “automatic differentiation” with the Laplace approximation for calculating the marginal likelihood. Automatic differentiation, which should not be confused with symbolic differentiation, is mostly unknown to statisticians, and hence basic ideas and results are reviewed. The computational performance of the approach is compared to that of existing mixed-model software on a suite of datasets selected from the mixed-model literature.