Methods for Default and Robust Bayesian Model Comparison: the Fractional Bayes Factor Approach

In the Bayesian approach to model selection and hypothesis testing, the Bayes factor plays a central role. However, the Bayes factor is very sensitive to prior distributions of parameters. This is a problem especially in the presence of weak prior information on the parameters of the models. The most radical consequence of this fact is that the Bayes factor is undetermined when improper priors are used. Nonetheless, extending the non‐informative approach of Bayesian analysis to model selection/testing procedures is important both from a theoretical and an applied viewpoint. The need to develop automatic and robust methods for model comparison has led to the introduction of several alternative Bayes factors. In this paper we review one of these methods: the fractional Bayes factor (O'Hagan, 1995). We discuss general properties of the method, such as consistency and coherence. Furthermore, in addition to the original, essentially asymptotic justifications of the fractional Bayes factor, we provide further finite‐sample motivations for its use. Connections and comparisons to other automatic methods are discussed and several issues of robustness with respect to priors and data are considered. Finally, we focus on some open problems in the fractional Bayes factor approach, and outline some possible answers and directions for future research. Dans I'approche Bayesienne relative a la selection d'un model et a la verification d'une hypothese, le facteur de Bayes joue une role fondamental. Toutefois le facteur de Bayes est tres sensible aux distributions a priori des parametres. Ceci constitue un probleme surtout en presence d'une faible information a priori en ce qui concerne les parametres des models. La consequence la plus radical de ce fait est que le facteur de Bayes est undetermine quand les distributions a priori non informatives sont utilisees. Cepandant, il est important d'elargir l'approche non informative de l'analyse Bayesienne a l'effet soit de determiner la selection d'un model que de verifier une hypothese. La necessite de developper des methodes automatiques et robustes pour la comparaison des models, a amenea l'introduction des plusieurs facteurs de Bayes alternatifs.Cette etude prend en consideration les resultats principaux relatifs a une de ces methodes, a savvoir le facteur de Bayes fractionnaire. Nous amalysons les caracteristique generales de cettemethode telles que sa consistance et sa coherence. De plus en sus des justifications asyntotiques donnees a l'origine au facteur fractionnaire de Bayes nous apportons d'autres raisons qui demontrent le bien fonde de4 son utilisation dans le domaine d'un echantillonage fini. Nous prenons aussien consideration par comparaison d'autres methodes automatiqueset nous examinations d'autres caracteristiques telles que la robustesse par rapport aux les distributions a priori et aux donnees. En conclusion, nous attirons l'attention sur certains problemes non encore resolus et proposons des solutions qui peuvent etre explorees d'avantage.

[1]  D. Pauler.  The Schwarz criterion and related methods for normal linear models , 1998 .

[2]  J. Berger,et al.  Testing Precise Hypotheses , 1987 .

[3]  Multiple Bayes Factors for Testing Hypotheses , 1995 .

[4]  Larry Wasserman,et al.  The conflict between improper priors and robustness , 1996 .

[5]  George E. P. Box,et al.  Sampling and Bayes' inference in scientific modelling and robustness , 1980 .

[6]  L. Wasserman,et al.  Computing Bayes Factors Using a Generalization of the Savage-Dickey Density Ratio , 1995 .

[7]  Julia Mortera,et al.  Robust Bayesian Hypothesis Testing in the Presence of Nuisance Parameters , 1994 .

[8]  A. O’Hagan,et al.  Properties of intrinsic and fractional Bayes factors , 1997 .

[9]  Fulvio Spezzaferri,et al.  A Predictive Model Selection Criterion , 1984 .

[10]  James O. Berger,et al.  On The Justification of Default and Intrinsic Bayes Factors , 1996 .

[11]  G. Casella,et al.  Reconciling Bayesian and Frequentist Evidence in the One-Sided Testing Problem , 1987 .

[12]  J. Berger,et al.  The Intrinsic Bayes Factor for Model Selection and Prediction , 1996 .

[13]  J. Berger,et al.  Testing a Point Null Hypothesis: The Irreconcilability of P Values and Evidence , 1987 .

[14]  A. F. Smith.  Present Position and Potential Developments: Some Personal Views Bayesian Statistics , 1984 .

[15]  J. Berger.  Statistical Decision Theory and Bayesian Analysis , 1988 .

[16]  F. Spezzaferri,et al.  Bayes Factors and Hierarchical Models , 1998 .

[17]  Luis R. Pericchi,et al.  On the robustness of the intrinsic Bayes factor for nested models , 1996 .

[18]  Takamitsu Sawa,et al.  Information criteria for discriminating among alternative regression models / BEBR No. 455 , 1978 .

[19]  Ward Edwards,et al.  Bayesian statistical inference for psychological research. , 1963 .

[20]  S. Geisser,et al.  A Predictive Approach to Model Selection , 1979 .

[21]  J. Berger,et al.  Ranges of posterior measures for priors with unimodal contaminations , 1989 .

[22]  A. P. Dawid,et al.  Present position and potential developments: some personal views , 1984 .

[23]  D. Spiegelhalter,et al.  Bayes Factors and Choice Criteria for Linear Models , 1980 .

[24]  F. Spezzaferri,et al.  Comparing hierarchical models using Bayes factor and fractional Bayes factor: a robust analysis , 1996 .

[25]  Fulvio Spezzaferri,et al.  Alternative Bayes factors for model selection , 1997 .

[26]  N. Draper,et al.  Applied Regression Analysis , 1966 .

[27]  Walter Racugno,et al.  An Intrinsic Limiting Procedure for Model Selection and Hypotheses Testing , 1998 .

[28]  D. Spiegelhalter,et al.  Bayes Factors for Linear and Log‐Linear Models with Vague Prior Information , 1982 .

[29]  Lea Petrella,et al.  Bayes factors at work in a challenging class of problems , 1998 .

[30]  Julia Mortera,et al.  Default Bayes Factors for Nonnested Hypothesis Testing , 1999 .

[31]  L. Wasserman,et al.  The Selection of Prior Distributions by Formal Rules , 1996 .