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.
A Note on Bayesian Modeling Specification of Censored Data in JAGS
Just Another吉布斯抽样（JAGS）是一个方便的工具，可以使用马尔可夫链蒙特卡罗来绘制贝叶斯建模的后验样本。然而，用于对删失数据建模的内置函数dinterval（）错误地指定了偏差函数的计算，这可能限制其用于执行基于似然的模型比较。为了建立一个自动的方法来指定正确的偏差函数在JAGS，我们提出了一个简单的替代建模策略，实现贝叶斯模型选择分析删失的结果。所提出的方法适用于广泛的数据类型，其中包括生存数据和许多其他的右，左和区间删失贝叶斯模型结构。
