Dilations and information flow axioms in categorical probability

TL;DR

This paper investigates the axioms of positivity and causality in Markov categories, revealing their structural implications, relationships, and limitations, especially in the context of probabilistic models and privacy considerations.

Contribution

It characterizes positivity in Markov categories, shows causality implies positivity, and explores the failure of positivity in quasi-Borel spaces, linking it to privacy.

Findings

Positivity is an intrinsic property of symmetric monoidal categories.

Causality implies positivity, but not vice versa.

Positivity fails in quasi-Borel spaces, indicating a privacy aspect.

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.