Partial Markov Categories

Abstract

We introduce partial Markov categories as a synthetic framework for synthetic probabilistic inference, blending the work of Cho and Jacobs, Fritz, and Golubtsov on Markov categories with the work of Cockett and Lack on cartesian restriction categories. We describe observations, Bayes' theorem, normalisation, and both Pearl's and Jeffrey's updates in purely categorical terms.

Figures (31)

• Figure 1: Copying and discarding equations.
• Figure 2: Axioms of comparators in a discrete cartesian restriction category.
• Figure 3: Conditionals split joint channels into a \ref{['linkMarginal']}$f$ and a conditional $g$.
• Figure 4: Set-up for the Three Prisoners Problem.
• Figure 5: Set-up for the Three Prisoners Problem, after the equality check.

Theorems & Definitions (85)

• Definition 2.1: Deterministic and total morphisms
• Definition 2.2: Marginals
• Definition 2.3: Conditional composition
• Remark 2.4: Type annotations
• Proposition 2.5
• proof
• Definition 2.6: Conditionals
• Definition 2.7: Markov category
• Definition 2.8: Almost-sure equality
• Proposition 2.9: Conditional of a deterministic morphism