Partial Markov Categories

2025-01-24

arXiv.org

AI Summary

This work addresses the lack of a unified semantic foundation for observation modeling, Bayesian updating, normalization, and both Pearl-style (causal intervention) and Jeffrey-style (soft evidence) conditioning in probabilistic reasoning. Methodologically, it introduces *partial Markov categories*—a novel categorical structure obtained by the first systematic integration of Markov categories with Cartesian restriction categories—enabling purely categorical, axiomatized characterizations of observations, Bayes’ theorem, normalization, and both update paradigms. The key contribution is the construction of the first compositional framework that uniformly supports both causal interventions (à la Pearl) and Jeffrey-style belief updates. Crucially, all central concepts—including conditioning, normalization, and update rules—emerge naturally from the categorical structure itself, without requiring external probabilistic interpretations. This provides a compositional, abstraction-layered semantic foundation for probabilistic programming languages and causal inference systems.

Technology Category

• Reasoning under Uncertainty: Probabilistic Programming
• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Action, Change, and Causality

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

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.

Problem

• Developing categorical framework for probabilistic inference
• Integrating Markov categories with restriction categories
• Describing Bayesian updates in categorical terms

Innovation

• Partial Markov categories framework
• Categorical probabilistic inference modeling
• Bayes theorem categorical implementation