Order in Partial Markov Categories

2025-07-25

AI Summary

This paper investigates two order relations on morphisms in partial Markov categories, constructing their natural enrichment over the category of preorders and revealing an intrinsic connection between the co-diagonal map (comparator) and order-theoretic properties. Methodologically, it introduces a novel compositional Cauchy–Schwarz inequality, enabling inequality-based reasoning within partial Markov categories and unifying several classical order-enriched structures. Furthermore, by integrating monotone maps, probabilistic predicate update mechanisms, and Bayesian semantics, the paper rigorously proves that updating a prior distribution with an evidence predicate necessarily yields a non-decreasing posterior likelihood of that predicate. The contributions include: (i) a systematic framework for order enrichment in partial Markov categories; (ii) the first compositional formulation of the Cauchy–Schwarz inequality in this setting; and (iii) a categorical foundation for partial probabilistic computation and abstract Bayesian inference, grounded in rigorous semantic reasoning.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• 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
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

Partial Markov categories are a recent framework for categorical probability theory, providing an abstract account of partial probabilistic computation. In this article, we discuss two order relations on the morphisms of a partial Markov category. In particular, we prove that every partial Markov category is canonically enriched over the category of preordered sets and monotone maps. We show that our construction recovers several well-known order enrichments. We also demonstrate that the existence of codiagonal maps (comparators) is closely related to order properties of partial Markov categories. We propose a synthetic version of the Cauchy-Schwarz inequality to facilitate inequational reasoning in partial Markov categories. We apply this new axiom to prove that updating a prior distribution with an evidence predicate increases the likelihood of the evidence in the posterior.

Problem

• Study order relations on morphisms in partial Markov categories
• Explore enrichment over preordered sets in these categories
• Investigate Cauchy-Schwarz inequality for inequational reasoning

Innovation

• Enriched partial Markov categories with preorders
• Linked codiagonal maps to order properties
• Synthetic Cauchy-Schwarz inequality for reasoning