Bayesian inference paradigms are regarded as powerful tools for solution of inverse problems. However, when applied to inverse problems in physical sciences, Bayesian formulations suffer from a number of inconsistencies that are often overlooked. A well known, but mostly neglected, difficulty is connected to the notion of conditional probability densities. Borel, and later Kolmogorov's (1933/1956), found that the traditional definition of conditional densities is incomplete: In different parameterizations it leads to different results. We will show an example where two apparently correct procedures applied to the same problem lead to two widely different results. Another type of inconsistency involves violation of causality. This problem is found in model selection strategies in Bayesian inversion, such as Hierarchical Bayes and Trans-Dimensional Inversion where so-called hyperparameters are included as variables to control either the number (or type) of unknowns, or the prior uncertainties on data or model parameters. For Hierarchical Bayes we demonstrate that the calculated 'prior' distributions of data or model parameters are not prior-, but posterior information. In fact, the calculated 'standard deviations' of the data are a measure of the inability of the forward function to model the data, rather than uncertainties of the data. For trans-dimensional inverse problems we show that the so-called evidence is, in fact, not a measure of the success of fitting the data for the given choice (or number) of parameters, as often claimed. We also find that the notion of Natural Parsimony is ill-defined, because of its dependence on the parameter prior. Based on this study, we find that careful rethinking of Bayesian inversion practices is required, with special emphasis on ways of avoiding the Borel-Kolmogorov inconsistency, and on the way we interpret model selection results.

翻译：贝叶斯推断范式被视为求解反问题的有力工具。然而，当应用于物理科学中的反问题时，贝叶斯公式存在诸多常被忽视的不一致性。一个众所周知但大多被忽略的难点与条件概率密度的概念相关。波莱尔以及后来的科尔莫戈罗夫（1933/1956）发现，条件密度的传统定义是不完备的：在不同的参数化方式下，该定义会导致不同的结果。本文将展示一个典型案例，其中两种表面正确的处理过程应用于同一问题，却得出迥异的结果。另一种不一致性涉及因果性的违反。这一问题在贝叶斯反演的模型选择策略中有所体现，例如层次贝叶斯和跨维度反演，其中所谓的超参数被作为变量引入，以控制未知量的数量（或类型），或数据及模型参数的先验不确定性。对于层次贝叶斯方法，我们证明计算得到的“数据或模型参数的’先验‘分布”实际并非先验信息，而是后验信息。事实上，计算得到的数据“标准差”反映的是正演函数对数据建模能力的不足，而非数据的不确定性。对于跨维反问题，我们显示所谓的证据实际上并非如通常所声称的那样，是衡量在给定参数选择（或数量）下数据拟合成功程度的指标。我们还发现“自然简约性”这一概念定义不清，因其依赖于参数的先验分布。基于本研究，我们认为需要谨慎反思贝叶斯反演实践，尤其应关注如何避免波莱尔-科尔莫戈罗夫不一致性，以及如何解读模型选择的结果。