一句话：以范畴论和 copartiality 形式化开放随机系统，并统一扩展高斯分布与线性关系。
论文研究 Jan Willems 风格的开放随机系统，其中不确定性分为概率波动与因根本信息缺失而产生的非确定性。作者用范畴论对开放随机系统进行形式化，核心是 copartiality 概念，用于刻画信息缺失如何沿系统传播，对应 Willems 工作中 sigma-代数的粗糙化。作为具体实例，论文研究扩展高斯分布，将高斯概率与非确定性结合，并证明其恰好对应 Willems 的高斯线性系统。作者同时给出测度论和抽象范畴论两种描述，从而严格刻画噪声物理定律、贝叶斯统计中的无信息先验等现象。扩展高斯映射范畴可视为高斯概率与线性关系的相互推广，连接范畴概率与信号流图等控制论思想。
An open stochastic system à la Jan Willems is a system affected by two qualitatively different kinds of uncertainty: one is probabilistic fluctuation, and the other one is nondeterminism caused by a fundamental lack of information. We present a formalization of open stochastic systems in the language of category theory. Central to this is the notion of copartiality which models how the lack of information propagates through a system (corresponding to the coarseness of sigma-algebras in Willems' work). As a concrete example, we study extended Gaussian distributions, which combine Gaussian probability with nondeterminism and correspond precisely to Willems' notion of Gaussian linear systems. We describe them both as measure-theoretic and abstract categorical entities, which enables us to rigorously describe a variety of phenomena like noisy physical laws and uninformative priors in Bayesian statistics. The category of extended Gaussian maps can be seen as a mutual generalization of Gaussian probability and linear relations, which connects the literature on categorical probability with ideas from control theory like signal-flow diagrams.
DOI 原文 · arXiv · PDF（开放获取） · DBLP
@article{abs-2403-03934,
title = {A Categorical Treatment of Open Linear Systems},
author = {Dario Stein and Richard Samuelson},
journal = {Logical Methods in Computer Science},
volume = {Volume 21, Issue 3},
year = {2025},
doi = {10.46298/lmcs-21(3:11)2025}
}