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Dilations and information flow axioms in categorical probability

Published online by Cambridge University Press:  25 October 2023

Tobias Fritz Open the ORCID record for Tobias Fritz [Opens in a new window] ,
Tomáš Gonda ,
Nicholas Gauguin Houghton-Larsen ,
Antonio Lorenzin ,
Paolo Perrone  and
Dario Stein
Tobias Fritz*
Affiliation:
University of Innsbruck, Innsbruck, Austria
Tomáš Gonda
Affiliation:
University of Innsbruck, Innsbruck, Austria
Nicholas Gauguin Houghton-Larsen
Affiliation:
Copenhagen, Denmark
Antonio Lorenzin
Affiliation:
University of Innsbruck, Innsbruck, Austria
Paolo Perrone
Affiliation:
University of Oxford, Oxford, UK
Dario Stein
Affiliation:
Radboud Universiteit Nijmegen, Nijmegen, Netherlands
*
Corresponding author: Tobias Fritz; Email: tobias.fritz@uibk.ac.at
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Abstract

We study the positivity and causality axioms for Markov categories as properties of dilations and information flow and also develop variations thereof for arbitrary semicartesian monoidal categories. These help us show that being a positive Markov category is merely an additional property of a symmetric monoidal category (rather than extra structure). We also characterize the positivity of representable Markov categories and prove that causality implies positivity, but not conversely. Finally, we note that positivity fails for quasi-Borel spaces and interpret this failure as a privacy property of probabilistic name generation.

Keywords

Categorical probabilityMarkov categorySemicartesian categoryInformation flowQuasi-Borel space

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Type
Paper
Information
Mathematical Structures in Computer Science , Volume 33 , Issue 10 , November 2023 , pp. 913 - 957
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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