A Category for Unifying Gaussian Probability and Nondeterminism

Authors Dario Stein, Richard Samuelson

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Author Details

Dario Stein
  • iHub, Radboud University Nijmegen, The Netherlands
Richard Samuelson
  • Humming Inc., Tacoma, WA, USA


It has been useful to discuss this work with many people. Particular thanks go to Tobias Fritz, Bart Jacobs, Dusko Pavlovic, Sam Staton and Alexander Terenin.

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Dario Stein and Richard Samuelson. A Category for Unifying Gaussian Probability and Nondeterminism. In 10th Conference on Algebra and Coalgebra in Computer Science (CALCO 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 270, pp. 13:1-13:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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 ⊆ 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Probabilistic computation
  • Theory of computation → Categorical semantics
  • Mathematics of computing → Probability and statistics
  • systems theory
  • hypergraph categories
  • Bayesian inference
  • category theory
  • Markov categories


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