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

Fulvio De Santis and Fulvio Spezzaferri

Abstract:

In the Bayesian approach to model selection and hypothesis testing, the Bayes factor, which is the ratio of posterior to prior odds of two models under comparison, plays a central role. However the Bayes factor, which is analytically equal to the ratio of the marginal distributions of the data under the two models, 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, defined only up to arbitrary constants, are used. Nonetheless, extending the non informative approach of Bayesian analysis to model selection/testing procedures is important from both a theoretical and an applied viewpoint. The need to develope 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. We then consider several issues of robustness with respect to priors and data. Finally, we focus on some open problems in the Fractional Bayes factor approach, and outline some possible answers and directions for future research. Keywords: Bayes Factors, Fractional Bayes Factors, Intrinsic Bayes Factors, Model Comparison, Robustness.

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