DATA-DRIVEN GIBBS SAMPLING

                      George Casella
                  Department of Biometrics
                    Cornell University

Computational methods such as the Gibbs sampler, and other MCMC
techniques, have greatly enhanced our ability to analyze complicated
models.  In particular, modeling with a hierarchical structure has become
quite common.  Such models often make it necessary for an experimenter to
specify the values of hyperparameters--parameters of distributions that
are deep in the hierarchy.  As experimenters (and most others) are
typically not versed in such an endeavor, there is little experience or
guidance on how to choose values for the hyperparameters (other than
invoking 'vagueness' or 'diffuseness', which may not be appropriate).
The obvious attack is to estimate the hyperparameters.

Although estimation of hyperparameters will typically
have little effect on the validity of the computational algorithm
employed, it can have great effect on the inference.  In particular, what
started off as a Bayes inference may end up as a frequentist inference
(all by itself!).  We explore these situations, giving some results on
the types of inference that are possible.  We also examine some
variations on the Gibbs samplers and the EM algorithm that allow
straightforward computation with estimated hyperparameters.