Sankhya: The Indian Journal of Statistics

1998, Volume 60, Series B, Pt. 1, 19--30

A NOTE ON FIRST-STAGE APPROXIMATION IN TWO-STAGE HIERARCHICAL MODELS

By

MICHAEL J. DANIELS
*Iowa State University, Ames*

and

ROBERT E. KASS,
* Carnegie Mellon University, Pittsburgh *

*SUMMARY.* 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).

*AMS (1991) subject classification.* 62C10.

*Key words and phrases. *Laplace's method, MCMC, importance sampling.