A note on the area under the likelihood and the fake evidence for model selection

TL;DR

This paper explores the use of improper priors in a specific model selection context, introducing the concept of 'fake evidences' as areas under the likelihood, and discusses their properties and limitations through theoretical and numerical analysis.

Contribution

It introduces the concept of 'fake evidences' for model selection with improper priors and analyzes their behavior, especially the limitations of using diffuse priors to approximate these quantities.

Findings

Improper priors can be used for 'fake evidences' in certain model selection scenarios.

Increasing the scale of a diffuse prior does not recover the area under the likelihood.

Numerical experiments confirm theoretical insights about 'fake evidences'.

Abstract

Improper priors are not allowed for the computation of the Bayesian evidence $Z=p({\bf y})$ (a.k.a., marginal likelihood), since in this case $Z$ is not completely specified due to an arbitrary constant involved in the computation. However, in this work, we remark that they can be employed in a specific type of model selection problem: when we have several (possibly infinite) models belonging to the same parametric family (i.e., for tuning parameters of a parametric model). However, the quantities involved in this type of selection cannot be considered as Bayesian evidences: we suggest to use the name ``fake evidences'' (or ``areas under the likelihood'' in the case of uniform improper priors). We also show that, in this model selection scenario, using a diffuse prior and increasing its scale parameter asymptotically to infinity, we cannot recover the value of the area under the likelihood, obtained with a uniform improper prior. We first discuss it from a general point of view. Then we provide, as an applicative example, all the details for Bayesian regression models with nonlinear bases, considering two cases: the use of a uniform improper prior and the use of a Gaussian prior, respectively. A numerical experiment is also provided confirming and checking all the previous statements.