Building on recent work in statistical science, the paper presents a theory for modelling natural phenomena that unifies physical and statistical paradigms based on the underlying principle that a model must be nondimensionalizable. After all, such phenomena cannot depend on how the experimenter chooses to assess them. Yet the model itself must be comprised of quantities that can be determined theoretically or empirically. Hence, the underlying principle requires that the model represents these natural processes correctly no matter what scales and units of measurement are selected. This goal was realized for physical modelling through the celebrated theories of Buckingham and Bridgman and for statistical modellers through the invariance principle of Hunt and Stein. Building on recent research in statistical science, the paper shows how the latter can embrace and extend the former. The invariance principle is extended to encompass the Bayesian paradigm, thereby enabling an assessment of model uncertainty. The paper covers topics not ordinarily seen in statistical science regarding dimensions, scales, and units of quantities in statistical modelling. It shows the special difficulties that can arise when models involve transcendental functions, such as the logarithm which is used e.g. in likelihood analysis and is a singularity in the family of Box-Cox family of transformations. Further, it demonstrates the importance of the scale of measurement, in particular how differently modellers must handle ratio- and interval-scales
翻译:本文件以统计科学的近期工作为基础,提出了一种对自然现象进行建模的理论,这些自然现象以模型必须不可扩展的基本原则为基础,将物理和统计范式统一起来。归根结底,这些现象不能取决于实验者如何选择评估这些现象。但模型本身必须包含在理论上或经验上可以确定的数量。因此,基本原则要求模型正确代表这些自然过程,无论在统计建模中选择何种尺度和单位。这一目标通过著名的巴克林汉姆和布林曼理论,以及通过亨特和施泰因的逆差原则为统计建模者而实现。在统计科学的最新研究基础上,论文表明后者能够接受和扩大前者。变量原则的范围扩大到包括巴伊西亚模式,从而能够评估模型的不确定性。本文涵盖了统计科学中通常没有看到的有关统计模型规模、尺度和数量单位等专题。它显示了当模型涉及超常性功能时可能出现的特殊困难,例如使用的对数值的对数分析,以及统计模型在模型中具有何种可能性,是不同规模的模型的超常识度,是不同规模的缩度。