一句话：从合适 Markov 范畴构造满足 Simpson 公理的样本空间范畴，并合成式建立概率层理论。
概率论存在两种高层图景：以随机变量为中心，以及以分布/概率通道（Markov 核）为中心。后者已由 Markov 范畴成功公理化并广泛推广，而随机变量图景的范畴语义仍不清晰。Simpson 的概率层最近提出，让随机变量等概率概念可在样本空间 site 上变化，并识别出条件独立等丰富结构。本文将该方向与 Markov 范畴的一般性和抽象性结合：证明对任意合适的 Markov 范畴，可定义满足 Simpson 公理的样本空间范畴，并在此框架中纯粹合成式地发展概率层理论；还从已知 Markov 范畴统一恢复 Simpson 的例子，并考虑进一步推广。
Two high-level "pictures" of probability theory have emerged: one that takes as central the notion of random variable, and one that focuses on distributions and probability channels (Markov kernels). While the channel-based picture has been successfully axiomatized, and widely generalized, using the notion of Markov category, the categorical semantics of the random variable picture remain less clear. Simpson’s probability sheaves are a recent approach, in which probabilistic concepts like random variables are allowed vary over a site of sample spaces. Simpson has identified rich structure on these sites, most notably an abstract notion of conditional independence, and given examples ranging from probability over databases to nominal sets.We aim bring this development together with the generality and abstraction of Markov categories: We show that for any suitable Markov category, a category of sample spaces can be defined which satisfies Simpson’s axioms, and that a theory of probability sheaves can be developed purely synthetically in this setting. We recover Simpson’s examples in a uniform fashion from well-known Markov categories, and consider further generalizations.
DOI 原文 ·
@inproceedings{paperbot8663,
title = {Random Variables, Conditional Independence and Categories of Abstract Sample Spaces},
author = {Dario Stein},
booktitle = {LICS},
year = {2025},
doi = {10.1109/lics65433.2025.00042}
}