A Categorical Treatment of Open Linear Systems
Abstract
An open stochastic system à 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 sigmaalgebras). 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 measuretheoretic 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 signalflow diagrams.
 Publication:

arXiv eprints
 Pub Date:
 March 2024
 DOI:
 10.48550/arXiv.2403.03934
 arXiv:
 arXiv:2403.03934
 Bibcode:
 2024arXiv240303934S
 Keywords:

 Computer Science  Logic in Computer Science;
 Mathematics  Category Theory;
 Mathematics  Probability;
 G.3;
 F.3.2;
 F.4.0