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

arXiv: https://arxiv.org/abs/2602.22965 · HTML: https://mlsanity.com/paper/2602.22965/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 28 references that are in the map.
- First posted on arXiv 26 Feb 2026.

## Builds on (most cited first, 7 of 7)

- [Marginal likelihood computation for model selection and hypothesis testing: an extensive review](https://mlsanity.com/paper/2005.08334/) (2020)
- [A Joint introduction to Gaussian Processes and Relevance Vector Machines with Connections to Kalman filtering and other Kernel Smoothers](https://mlsanity.com/paper/2009.09217/) (2020)
- [Deep Importance Sampling based on Regression for Model Inversion and Emulation](https://mlsanity.com/paper/2010.10346/) (2020)
- [Automatic tempered posterior distributions for Bayesian inversion problems](https://mlsanity.com/paper/2107.11614/) (2021)
- [Recursive Pathways to Marginal Likelihood Estimation with Prior-Sensitivity Analysis](https://mlsanity.com/paper/1301.6450/) (2013)
- [On the safe use of prior densities for Bayesian model selection](https://mlsanity.com/paper/2206.05210/) (2022)
- [Compressed particle methods for expensive models with application in Astronomy and Remote Sensing](https://mlsanity.com/paper/2107.08465/) (2021)

## Abstract (arXiv)

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.