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

• URL: http://arxiv.org/abs/2602.22965v1
• Date: Thu, 26 Feb 2026 13:01:50 GMT
• Title: A note on the area under the likelihood and the fake evidence for model selection
• Authors: L. Martino, F. Llorente,
• Abstract summary: Improper priors are not allowed for the computation of the Bayesian evidence $Z=p(bf y)$ (a.k.a., marginal likelihood)<n>We show that they can be employed in a specific type of model selection problem.<n>A numerical experiment is also provided confirming and checking all the previous statements.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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