98
J. A. Cano, D. Salmer´on
sample is a sample of minimal size for which the posterior
π
N
i
(
θ
i

x
) is proper,
i
= 1
,
2. The main concern when using this methodology is the choice of
m
∗
(
x
)
which is difficult to assess, mostly when comparing nonnested models.
One more alternative solution consists of producing priors satisfying some
type of objective criteria. Among the employed criteria some of them are related
with consistency in different ways, for instance, chosing the true model as the
sample size goes to infinity that is called model selection consistency, see Bayarri
et al
., 2012, for other consistency criteria. Another, important criterion is exact
predictive matching, that is fulfilled when the priors for the two models under
comparison
{
π
1
(
θ
1
)
, π
2
(
θ
2
)
}
are such that their corresponding predictive margi
nals
m
1
(
x
) and
m
2
(
x
) are equal for every imaginary minimal training sample
x
.
In section 2 we introduce integral priors mentioning some of their good pro
perties and very quickly we go down to its core showing how they work in the
theoretical and in the practical way. In section 3 we review a simple case, that is
testing a normal mean with known variance. The case was dealt with in Cano
et
al
.,2016, where it was treated like a testing problem (not an estimation problem)
using improper priors and therefore giving rise to the problem of undefined Bayes
factors. Here we review how numerical computation can be carried out through
this simple example and that the approximated Bayes factors obtained are very
good approximations, in fact this example can be considered as a guide to use
integral priors to be applied later to solve more complex problems. In addition
the nice property of the integral prior for the complex model of concentrating
mass in the null when comparing nested models is exhibited.
In section 4 we review more complex applications that have been carried out
using the integral priors methodology, like testing in the scenarios of a Cauchy
distribution and binomial regression models.
Section 5 is devoted to summarize the application of integral priors to the
one way random effects model developed in Cano
et al
, 2007a, 2007b, and we
take advantage of these papers to improve the presentation of some results in
section 4 of Cano
et al
, 2007a.
Finally, in section 6 we present some relevant conclusions and outline onco
ming research.
2. Introducing the integral priors
Integral priors are a new methodology to deal with Bayesian model selection
problems considered as a generalization of hypothesis testing problems, since
they allow to compare nonnested models. The integral priors were proposed in
Cano
et al
., 2008, where under mild assumptions it was proved that they are
unique up to a multiplicative constant that is canceled out in the computation
of the Bayes factor.