  Model Selections in Bayesian Statistics are primarily made with statistics
known as Bayes Factors, which are directly related to Posterior Probabilities
of models. Bayes Factors require a careful assessment of prior distributions as
in the Intrinsic Priors of Berger and Pericchi (1996a) and integration over the
parameter space, which may be highly dimensional. Recently researchers have
been proposing alternatives to Bayes Factors that require neither integration
nor specification of priors. These developments are still in a very early stage
and are known as Prior-free Bayes Factors, Cross-Validation Bayes Factors
(CVBF), and Bayesian "Stacking." This kind of method and Intrinsic Bayes Factor
(IBF) both avoid the specification of prior. However, this Prior-free Bayes
factor might need a careful choice of a training sample size. In this article,
a way of choosing training sample sizes for the Prior-free Bayes factor based
on Geometric Intrinsic Bayes Factors (GIBFs) is proposed and studied. We
present essential examples with a different number of parameters and study the
statistical behavior both numerically and theoretically to explain the ideas
for choosing a feasible training sample size for Prior-free Bayes Factors. We
put forward the "Bridge Rule" as an assignment of a training sample size for
CVBF's that makes them close to Geometric IBFs. We conclude that even though
tractable Geometric IBFs are preferable, CVBF's, using the Bridge Rule, are
useful and economical approximations to Bayes Factors.
