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Bolet´ın de Estad´ıstica e Investigaci´on Operativa

Vol. 32, No. 2, Julio 2016, pp. 96-111


A Review of the Developments on Integral Priors for

Bayesian Model Selection?

Juan A. Cano

Departamento de Estad´ıstica e Investigaci´on Operativa, Universidad de Murcia


Diego Salmer´on

Servicio de Epidemiolog´ıa, Consejer´ıa de Sanidad, IMIB-Arrixaca, Murcia

CIBER Epidemiolog´ıa y Salud P´ublica (CIBERESP)

Departamento de Ciencias Sociosanitarias, Universidad de Murcia



For the sake of objectivity it is a common practice in Bayesian model

selection using default priors. However, these priors are usually improper

yielding indeterminate Bayes factors that preclude the comparison of the

models. Because of this some approaches have been proposed to obtain

more refined default prior distributions avoiding the indetermination of

their associated Bayes factors. Among these approaches, a special mention

is deserved for the intrinsic priors that were introduced in Berger and

Pericchi, 1996. Another important development, the expected posterior

priors, appeared in P´erez and Berger, 2002. A special mention is also due

to the criteria based priors, a summary of which appears in Bayarri

et al



Here, we mainly focus on the integral priors, that were presented in the

germinal paper Cano

et al

., 2008, comparing them with the priors above

mentioned. These integral priors have been further developed in Cano

et al

., 2007a, and Cano

et al

., 2007b, where they were used to analyze

the random effects model, Cano and Salmer´on, 2013, where an extension

was introduced to deal with more sophisticated problems, and applied to

binomial regression models in Salmer´on

et al

., 2015. Cano

et al

., 2016, was

devoted to present a methodological introduction that could be useful as

a user guide for possible practitioners.

This review paper is intended to be a presentation of the state of the

art regarding the developments of the integral priors. One of the main

advantages of this methodology is that it can be applied to compare both


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