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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 non-nested 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 non-nested 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.