Dilations and information flow axioms in categorical probability
Tobias Fritz, Tomáš Gonda, Nicholas Gauguin Houghton-Larsen, Antonio Lorenzin, Paolo Perrone, Dario Stein
TL;DR
The paper addresses how positivity and causality axioms constrain probabilistic reasoning in Markov categories, establishing that causality implies positivity but not conversely, and showing that positivity is a structural property of the underlying symmetric monoidal category. It develops reformulations in terms of determinstic marginal independence and dilations, provides a representable-category criterion via strongly affine monads, and extends the analysis to semicartesian categories using dilations, including an abstract, dilationally grounded characterization of positive Markov categories. It also presents a counterexample in quasi-Borel spaces (QBStoch) where positivity fails, interpreted through the privacy equation and fresh-name generation, linking information flow with privacy. The work thus unifies categorical probability through dilations, semicartesian frameworks, and information-flow axioms, with implications for probabilistic programming and privacy-sensitive reasoning across models.
Abstract
We study the positivity and causality axioms for Markov categories as properties of dilations and information flow in Markov categories, and in 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.