Problem
Research questions and friction points this paper is trying to address.
Study positivity and causality axioms in Markov categories
Characterize positivity in representable Markov categories
Interpret positivity failure as privacy property
Dilations and information flow axioms in categorical probability
🤖 AI Summary
This paper investigates the fundamental relationship between the axioms of “positivity” and “causality” within Markov categories, particularly in the context of morphism extension and information flow. Method: By embedding both axioms into a unified framework of semi-Cartesian symmetric monoidal categories equipped with a symmetric monad, the authors develop a categorical semantics for probabilistic reasoning. Contribution/Results: The work establishes, for the first time, that positivity is an intrinsic property of symmetric monoidal categories—not additional structure—while causality strictly implies positivity, but not conversely. It fully characterizes necessary and sufficient conditions for positivity in representable Markov categories. Furthermore, it demonstrates that the failure of positivity in the quasi-Borel category is not a deficiency but reflects a privacy-preserving feature inherent to probabilistic naming generation. The study thereby clarifies the logical hierarchy and semantic distinctions among probabilistic axioms across diverse models, laying a rigorous categorical foundation for positivity and causality.
📝 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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Study positivity and causality axioms in Markov categories
Characterize positivity of representable Markov categories
Interpret positivity failure as privacy property
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