发表机构
Politecnico di Torino（都灵理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明统计模型是设计范畴到Stoch范畴函子间的自然变换，提出有意义性与相干性概念，并揭示岭回归作为高斯线性模型关于高斯先验的贝叶斯反演。
AI 中文摘要
我们证明了McCullagh意义上的统计模型，以Brøns给出的形式，是从设计范畴到Giry单子的Kleisli范畴Stoch的两个函子之间的自然变换，前提是其分量在参数上是可测的。对于有限模型，该条件为空。设计索引的量是定义在参数对象上的Stoch态射族，如果它是自然的，则称为有意义的。我们证明了Tjur准则，该准则施加于由带重数的有限样本索引的参数函数上，强制以支撑集为索引，然后与插入上的自然性一致。对于有限设计，我们证明了一个量可以在给定的修正类内被修正为自然的，当且仅当该类在相对于该类的Baues-Wirsching复形的第一上同调群中消失，而其在绝对群中的像始终为零。在单向布局中，边际离散度不是有意义的，而组内离散度是保持合并设计不变的唯一修正。先验是参数对象上的状态族，如果它在某类设计态射上是自然的，则称为在该类上相干。我们证明了在合并处的相干性将先验限制在相应参数映射的像内，插入上的相干性即Kolmogorov一致性，而注入上的相干性则添加了范畴de Finetti定理所假设的可交换性。在有限单向方案中，相干先验构成已知维数的多面体。Jeffreys一般规则的类比不是相干的，而位置-尺度族的类比是相干的。最后，我们证明了岭回归是关于高斯先验的高斯线性模型的贝叶斯反演，该先验在插入上是相干的，而在注入上永不相干。
英文摘要
We show that a statistical model in the sense of McCullagh, in the form given by Brøns, is a natural transformation between two functors from the category of designs to the Kleisli category Stoch of the Giry monad, provided that its components are measurable in the parameter. The condition is empty for finite models. A design-indexed quantity is a family of morphisms of Stoch defined on the parameter objects, called meaningful if it is natural. We prove that Tjur's criterion, imposed on parameter functions indexed by finite samples with multiplicities, forces the indexing by the support and then coincides with naturality over the insertions. For finite designs we show that a quantity can be corrected to a natural one within a given class of corrections if and only if a class vanishes in the first cohomology group of a Baues-Wirsching complex relative to that class, while its image in the absolute group is always zero. In the one-way layout, marginal dispersion is not meaningful, and within-group dispersion is the unique correction that leaves the merged design unchanged. A prior is a family of states on the parameter objects, called coherent over a class of design morphisms if it is natural over that class. We show that coherence at a merge confines the prior to the image of the corresponding parameter map, that coherence over the insertions is Kolmogorov consistency, and that coherence over the injections adds the exchangeability assumed by the categorical de Finetti theorem. In the finite one-way scheme, the coherent priors form polytopes of known dimension. The analogue of Jeffreys' general rule is not coherent, while the analogue for location-scale families is. Finally, we show that ridge regression is the Bayesian inversion of the Gaussian linear model with respect to a Gaussian prior, which is coherent over the insertions and never over the injections.
Comments39 pages. v3: related work corrected (Fritz 2020, Section 7 and Problem 11.9); Remarks 4.6 and 8.6 extended accordingly; numbering unchanged. Computational supplement: https://github.com/Vaxx66/calamus, release v36
