  In Bayesian hypothesis testing and model selection, prior distributions must
be chosen carefully. For example, setting arbitrarily large prior scales for
location parameters, which is common practice in estimation problems, can lead
to undesirable behavior in testing (Lindley's paradox). We study the properties
of some restricted type II maximum likelihood (type II ML) priors on regression
coefficients. In type II ML, hyperparameters are "estimated" by maximizing the
marginal likelihood of a model. In this article, we define priors by estimating
their variances or covariance matrices, adding restrictions which ensure that
the resulting priors are at least as vague as conventional proper priors for
model uncertainty. We find that these type II ML priors typically yield results
that are close to answers obtained with the Bayesian Information Criterion
(BIC).
