## A Note on First-Stage Approximation In Two-Stage Hierarchical Models

September, 1997

Tech Report

### Author(s)

Michael J. Daniels and Robert E. Kass

### Abstract

We consider approximations to two-stage hierarchical models in which the second stage uses a Normal distribution to model the variation of the first-stage parameters. If we replace the first stage of the model with a Normal distribution based on first-stage maximum likelihood estimation, we obtain an alternative two-stage model that approximates the original model while allowing posterior simulation to become easy and efficient. We note that the MLE-based Normal approximation is not quite a special case of Laplace's method, but it does produce the same accuracy as Laplace's method in approximating the posterior of the second-stage parameters. In a previous paper we showed how draws from such approximate posteriors may be reweighted to produce importance samples from the original posterior. Here we show how the method extends to mixed models, and hierarchical nonlinear models. We demonstrate the possible utility of this kind of scheme by easily obtaining posterior inferences (without special-purpose MCMC code) for a model that could not, at the time of our writing, be fit by BUGS (Spiegelhalter, Best, Gilks, Inskip, 1996).