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.

翻译：在计算贝叶斯证据 $Z=p({\bf y})$（亦称边缘似然）时，不允许使用非正常先验，因为在这种情况下，由于计算中涉及任意常数，$Z$ 无法被完全确定。然而，本工作中我们指出，在特定类型的模型选择问题中可以使用非正常先验：即当我们有多个（可能无限个）属于同一参数族（例如用于调整参数模型的超参数）的模型时。然而，此类选择问题中涉及的量不能被视作贝叶斯证据：我们建议使用“伪证据”这一名称（对于均匀非正常先验的情况，可称为“似然函数下方面积”）。我们还证明，在此类模型选择场景中，若使用扩散先验并将其尺度参数渐近增大至无穷大，我们无法恢复使用均匀非正常先验所得的似然函数下方面积值。我们首先从一般性角度进行讨论，随后以非线性基函数的贝叶斯回归模型作为应用示例，分别考虑使用均匀非正常先验与高斯先验两种情况，并提供了数值实验以验证和检验所有前述结论。