  The application of Bayesian inference for the purpose of model selection is
very popular nowadays. In this framework, models are compared through their
marginal likelihoods, or their quotients, called Bayes factors. However,
marginal likelihoods depends on the prior choice. For model selection, even
diffuse priors can be actually very informative, unlike for the parameter
estimation problem. Furthermore, when the prior is improper, the marginal
likelihood of the corresponding model is undetermined. In this work, we discuss
the issue of prior sensitivity of the marginal likelihood and its role in model
selection. We also comment on the use of uninformative priors, which are very
common choices in practice. Several practical suggestions are discussed and
many possible solutions, proposed in the literature, to design objective priors
for model selection are described. Some of them also allow the use of improper
priors. The connection between the marginal likelihood approach and the
well-known information criteria is also presented. We describe the main issues
and possible solutions by illustrative numerical examples, providing also some
related code. One of them involving a real-world application on exoplanet
detection.
