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

Antonio Lorenzin, Fabio Zanasi

TL;DR

The paper develops a categorical, syntax-then-semantics framework for Bayesian and Markov networks, representing them as functors from graph-like syntax to probabilistic semantics. It defines moralisation and triangulation as functors connecting BN and MN, and decomposes triangulation into syntactic and semantic stages using chordal and hypergraph structures. Irredundant networks are highlighted to ensure proper functoriality, and the variable elimination algorithm is interpreted as a semantic functor VE(-) that interacts cleanly with triangulation. The work demonstrates a modular, compositional approach to PGMs, clarifying the separation of syntactic structure and probabilistic content and enabling principled reasoning about transformations between directed and undirected graphical models. A program of future work points toward broader PGMs (Gaussian, factor graphs) and a deeper, algebraic treatment of inference algorithms in a fully categorical setting.

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.

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

TL;DR

The paper develops a categorical, syntax-then-semantics framework for Bayesian and Markov networks, representing them as functors from graph-like syntax to probabilistic semantics. It defines moralisation and triangulation as functors connecting BN and MN, and decomposes triangulation into syntactic and semantic stages using chordal and hypergraph structures. Irredundant networks are highlighted to ensure proper functoriality, and the variable elimination algorithm is interpreted as a semantic functor VE(-) that interacts cleanly with triangulation. The work demonstrates a modular, compositional approach to PGMs, clarifying the separation of syntactic structure and probabilistic content and enabling principled reasoning about transformations between directed and undirected graphical models. A program of future work points toward broader PGMs (Gaussian, factor graphs) and a deeper, algebraic treatment of inference algorithms in a fully categorical setting.

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.

Paper Structure

This paper contains 14 sections, 25 theorems, 43 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Conference version
  3. Outline
  4. Acknowledgements
  5. Copy-Delete and Hypergraph Categories
  6. Probabilistic Graphical Models as Functors
  7. Syntax Functoriality
  8. Irredundant Networks via Functorial Factorisation
  9. Morphisms Between Networks
  10. Moralisation as a Functor
  11. Triangulation as a Functor
  12. Variable Elimination as a Functor
  13. Functorial Interplay
  14. Conclusions

Key Result

Proposition 3.4

Let $\mathcal{G}$ be an ordered DAG. Bayesian networks over $\mathcal{G}$ are in bijective correspondence with CD-functors $\mathsf{CDSyn}_{\mathcal{G}}\to \mathsf{FinStoch}$.

Figures (5)

  • Figure 1: A Bayesian network and the string diagram in $\mathsf{FinStoch}$ representing it.
  • Figure 2: An undirected graph and the string diagram representing its unnormalised distribution. Although more graphically complex, the string diagram makes all contributing factors explicit. This approach is also commonly reflected in the theory of PGMs through the use of factor graphs, which are more descriptive.
  • Figure 3: Examples of the contravariant action of the functor of \ref{['thm:graphhom-cd']} on morphisms $\mathcal{G} \to \mathcal{G}'$ (top), resulting in CD-functors (bottom), of which we describe the action on generators of $\mathsf{CDSyn}_{\mathcal{G}}$.
  • Figure 4: Examples of the contravariant action of the functor of \ref{['thm:graphhom-hyp']} on morphisms $\mathcal{H} \to \mathcal{H}'$ (top), resulting in hypergraph functors (bottom), of which we describe the action on generators of $\mathsf{HSyn}_{\mathcal{H}'}$.
  • Figure 5: A visual example of cluster graphs and junction trees.

Theorems & Definitions (102)

  • Definition 2.2
  • Example 2.4
  • Example 2.5: Free CD-categories
  • Definition 2.6
  • Definition 2.8
  • Example 2.11
  • Example 2.12
  • Remark 2.13: The Normalisation Cospan
  • Definition 2.14: Tentative
  • Example 2.16: Free Hypergraph Categories
  • ...and 92 more