  The goal of this paper is to compare several widely used Bayesian model
selection methods in practical model selection problems, highlight their
differences and give recommendations about the preferred approaches. We focus
on the variable subset selection for regression and classification and perform
several numerical experiments using both simulated and real world data. The
results show that the optimization of a utility estimate such as the
cross-validation (CV) score is liable to finding overfitted models due to
relatively high variance in the utility estimates when the data is scarce. This
can also lead to substantial selection induced bias and optimism in the
performance evaluation for the selected model. From a predictive viewpoint,
best results are obtained by accounting for model uncertainty by forming the
full encompassing model, such as the Bayesian model averaging solution over the
candidate models. If the encompassing model is too complex, it can be robustly
simplified by the projection method, in which the information of the full model
is projected onto the submodels. This approach is substantially less prone to
overfitting than selection based on CV-score. Overall, the projection method
appears to outperform also the maximum a posteriori model and the selection of
the most probable variables. The study also demonstrates that the model
selection can greatly benefit from using cross-validation outside the searching
process both for guiding the model size selection and assessing the predictive
performance of the finally selected model.
