On the safe use of prior densities for Bayesian model selection

TL;DR

This paper examines the sensitivity of Bayesian model selection to prior choices, discusses the use of uninformative and improper priors, and offers practical solutions and insights, including real-world applications.

Contribution

It provides a comprehensive analysis of prior sensitivity in Bayesian model selection and reviews methods for designing objective priors, including handling improper priors.

Findings

Prior choice significantly influences marginal likelihoods in model selection.

Uninformative priors can be informative in model comparison.

Several practical solutions for objective priors are discussed.

Abstract

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.