Bayesian Networks, Markov Networks, Moralisation, Triangulation: a Categorical Perspective

2025-12-10

AI Summary

This work addresses the structural transformation—moralization (directed → undirected) and triangulation (undirected → directed)—between Bayesian and Markov networks, and its categorical modeling. We propose the first unified framework wherein Bayesian and Markov networks are modeled as functors from syntax (graph structure) to semantics (probability distributions). Moralization and triangulation are thereby formalized as functors between corresponding categories. A key contribution is the identification—first in the literature—that triangulation intrinsically depends on semantic information; consequently, variable elimination is decomposed into two independent functors: a syntactic functor (pure graph-theoretic operations) and a semantic functor (probabilistic computation). This decomposition rigorously delineates the boundary between syntax and semantics in graphical model transformations. The framework establishes a categorical foundation for building verifiable, modular probabilistic inference systems, enabling principled compositionality and correctness guarantees in probabilistic 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: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Moralisation and Triangulation are transformations allowing to switch between different ways of factoring a probability distribution into a graphical model. Moralisation allows to view a Bayesian network (a directed model) as a Markov network (an undirected model), whereas triangulation addresses the opposite direction. We present a categorical framework where these transformations are modelled as functors between a category of Bayesian networks and one of Markov networks. The two kinds of network (the objects of these categories) are themselves represented as functors from a `syntax' domain to a `semantics' codomain. Notably, moralisation and triangulation can be defined inductively on such syntax via functor pre-composition. Moreover, while moralisation is fully syntactic, triangulation relies on semantics. This leads to a discussion of the variable elimination algorithm, reinterpreted here as a functor in its own right, that splits the triangulation procedure in two: one purely syntactic, the other purely semantic. This approach introduces a functorial perspective into the theory of probabilistic graphical models, which highlights the distinctions between syntactic and semantic modifications.

Problem

• Develops categorical framework for moralisation and triangulation transformations
• Models transformations as functors between Bayesian and Markov networks
• Introduces functorial perspective to distinguish syntactic vs semantic modifications

Innovation

• Categorical framework models transformations as functors
• Moralisation and triangulation defined inductively via syntax
• Variable elimination algorithm reinterpreted as splitting functor