Statistical models as natural transformations: meaningfulness, coherence and priors as states in Markov categories
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作为自然变换的统计模型：马尔可夫范畴中的有意义性、协调性与先验状态
- 研究问题
- 统计模型、设计索引的统计量和先验能否由自然变换统一表达，并用这一框架刻画统计量的有意义性及跨设计先验的协调性？
- 方法与思路
- 作者以设计范畴到随机映射范畴的函子和自然变换建模，并借助相对上同调讨论统计量修正，以状态族描述先验协调性。
- 材料陈述的贡献
- 摘要称，论文连接自然性、Tjur 准则与先验一致性，给出有限设计下的修正刻画，并分析单因素设计及岭回归的例子。
- 阅读关注点
- 应核对参数可测性条件、合并与各类嵌入映射的区别、相对上同调刻画的适用范围，以及先验协调性结论所依赖的设计类别。
- 阅读边界
- 这是 arXiv 预印本，不能据此认定已同行评审或期刊接收。摘要中的范畴论结论涉及特定条件，不能外推为所有模型的性质。
原文摘要
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.