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A Review of the Developments on Integral Priors for Bayesian Model Selection

97

nested and non-nested 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

)

2

c

1

!

f

1

(

x

|

θ

1

)

h

1

(

θ

1

)

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 non-nested 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