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

L. Martino () and F. Llorente
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L. Martino: Universita’ di Catania
F. Llorente: Stony Brook University

Computational Statistics, 2025, vol. 40, issue 8, No 26, 4799-4824

Abstract: Abstract Improper priors are not allowed for the computation of the Bayesian evidence $$Z=p(\textbf{y})$$ (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.

Keywords: Bayesian evidence; Marginal likelihood; Improper prior; Diffuse prior (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s00180-025-01641-2

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