A Categorical Treatment of Open Linear Systems
<|reference_start|>A Categorical Treatment of Open Linear Systems: 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.<|reference_end|>
@article{stein2024a,
    title={A Categorical Treatment of Open Linear Systems},
    author={Dario Stein, Richard Samuelson},
    journal={arXiv preprint arXiv:2403.03934},
    year={2024},
    archivePrefix={arXiv},
    eprint={2403.03934},
    primaryClass={cs.LO math.CT math.PR}
}