A Review of the Developments on Integral Priors for Bayesian Model Selection
97
nested and nonnested models. Another one is that integral priors are
invariant
σ
finite measures for two parallel Markov chains which simulation
very often can be carried out easily and therefore these Markov chains can
be used to approximate the corresponding Bayes factor.
Keywords:
Objective Bayesian model selection, Intrinsic priors, Ex
pected posterior priors, Criteria based priors, Integral priors.
AMS Subject classifications:
62F15.
1. Preview
Selecting prior distributions for Bayesian estimation and model selection is
sues is an old problem for which some solutions have been proposed in the last
decades. Default priors have preferably been used but usually they are impro
per prior distributions
π
N
(
θ
) =
ch
(
θ
), where
h
(
θ
) is a function whose integral
diverges, and the constant
c >
0 is arbitrary. In estimation this is not a problem
since the posterior does not depend on
c
. However, in model selection problems
when we have two models,
M
i
:
x
∼
f
i
(
x

θ
i
),
i
= 1
,
2, and default priors
π
N
i
(
θ
i
) =
c
i
h
i
(
θ
i
),
i
= 1
,
2, the Bayes factor
B
N
21
(
x
) =
m
N
2
(
x
)
m
N
1
(
x
)
=
c
2
!
f
2
(
x

θ
2
)
h
2
(
θ
2
)
dθ
2
c
1
!
f
1
(
x

θ
1
)
h
1
(
θ
1
)
dθ
1
,
depends on the arbitrary ratio
c
2
/c
1
.
Since we are dealing with model selection
a problem arises that needs to be solved: the indetermination of the ratio
c
2
/c
1
.
The proposed procedure consists in adjusting these default priors to produce
priors that avoid the indetermination problem.
A solution for this problem was introduced in Berger and Pericchi, 1996, it
consists of using intrinsic priors that are solutions to a system of two functional
equations. Nevertheless, very often intrinsic priors are not unique, for instance
when model
M
1
is nested in model
M
2
, the system of functional equations re
duces to a single equation with two
incognita
that usually has many solutions.
Likewise, in the nonnested case the class of intrinsic priors may be very large.
For instance, in Cano
et al
., 2004, it is shown that any couple of equal priors
are intrinsic when comparing the double exponential versus the normal location
models.
The expected posterior priors were stated in P´erez and Berger, 2002, as anot
her solution for the problem of the indetermination. In this article the authors
propose objective priors defined as expected posteriors under some common pre
dictive marginal
m
∗
(
x
) suitably chosen, that is,
π
∗
i
(
θ
i
) =
"
π
N
i
(
θ
i

x
)
m
∗
(
x
)
dx,
(1.1)
where
x
is an imaginary minimal training sample. Note that a minimal training