667

# 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.
*
*

Here is the full postscript text for this
technical report. It is 743224 bytes long.