An open stochastic system \`a la Willems is a system affected two qualitatively different kinds of uncertainty -- one is probabilistic fluctuation, and the other one is nondeterminism caused by lack of information. We give a formalization of open stochastic systems in the language of category theory. A new construction, which we term copartiality, is needed to model the propagating lack of information (which corresponds to varying sigma-algebras). As a concrete example, we discuss 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.

翻译：遵循Willems定义的开放随机系统受两种性质不同的不确定性影响——其一是概率波动，其二是由信息缺失产生的非确定性。我们采用范畴论语言对开放随机系统进行了形式化描述。为刻画传播性信息缺失（对应可变σ代数），我们提出了称为"共偏序性"的新构造。作为具体案例，我们讨论了扩展高斯分布——该分布将高斯概率与非确定性相结合，恰好对应Willems提出的高斯线性系统概念。我们从测度论与抽象范畴论两个层面对其进行描述，从而能够严谨地解释含噪物理定律、贝叶斯统计中的无信息先验等多种现象。扩展高斯映射范畴可视为高斯概率与线性关系的双向推广，这为将范畴概率文献与信号流图等控制论思想建立关联提供了桥梁。