Title:
Objective Bayesian Analysis for the Multivariate Normal Model
Abstract:
Objective Bayesian inference for the multivariate normal distribution is
illustrated, using different types of formal objective priors (Jeffreys,
invariant, reference and matching), different modes of inference
(Bayesian and frequentist), and different criteria involved in selecting
optimal objective priors (ease of computation, frequentist performance,
marginalization paradoxes, and decision-theoretic evaluation).
In the course of the investigation of the bivariate normal model, a
variety of surprising results were found,
including the availability of objective priors that yield exact
frequentist inferences for many functions of the bivariate normal
parameters, such as the correlation coefficient. Certain of these
results are generalized to the multivariate normal situation.
The prior that most frequently yields exact frequentist inference is the
right-haar prior, which unfortunately is not unique. Two natural
proposals are studied for dealing with this non-uniqueness: first,
mixing over the right-haar priors; second, choosing the `empirical
Bayes' right-haar prior, that which maximizes the marginal likelihood of
the data. Quite surprisingly, we show that neither of these
possibilities yields a good solution. This is disturbing and sobering.
It is yet another indication that improper priors do not behave as do
proper priors, and that it can be dangerous to apply `understandings'
from the world of proper priors to the world of improper priors.
Jointly with James O. Berger of Duke Univerty