We introduce categories of extended Gaussian maps and Gaussian relations which unify Gaussian probability distributions with relational nondeterminism in the form of linear relations. Both have crucial and well-understood applications in statistics, engineering, and control theory, but combining them in a single formalism is challenging. It enables us to rigorously describe a variety of phenomena like noisy physical laws, Willems' theory of open systems and uninformative priors in Bayesian statistics. The core idea is to formally admit vector subspaces $D \subseteq X$ as generalized uniform probability distribution. Our formalism represents a first bridge between the literature on categorical systems theory (signal-flow diagrams, linear relations, hypergraph categories) and notions of probability theory.

翻译：我们引入了扩展高斯映射与高斯关系范畴，该范畴以线性关系的形式统一了高斯概率分布与关系型非确定性。两者在统计学、工程学和控制理论中均具有关键且已充分理解的应用，但将它们整合至单一形式体系极具挑战性。这使我们能够严谨描述多种现象，如含噪声物理定律、Willems 开放系统理论以及贝叶斯统计中的无信息先验。核心思想是将向量子空间 $D \subseteq X$ 形式化地视为广义均匀概率分布。该形式体系首次在范畴系统理论文献（信号流图、线性关系、超图范畴）与概率论概念之间建立了桥梁。