# An Algebraic Approach to Moralisation and Triangulation of Probabilistic Graphical Models

arXiv: https://arxiv.org/abs/2503.11820 · HTML: https://mlsanity.com/paper/2503.11820/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 24 references that are in the map.
- First posted on arXiv 14 Mar 2025.

## Builds on (most cited first, 11 of 11)

- [A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics](https://mlsanity.com/paper/1908.07021/) (2019)
- [Category Theory in Machine Learning](https://mlsanity.com/paper/2106.07032/) (2021)
- [Causal Inference by String Diagram Surgery](https://mlsanity.com/paper/1811.08338/) (2018)
- [Causal models in string diagrams](https://mlsanity.com/paper/2304.07638/) (2023)
- [An Introduction to String Diagrams for Computer Scientists](https://mlsanity.com/paper/2305.08768/) (2023)
- [Hypergraph Categories](https://mlsanity.com/paper/1806.08304/) (2018)
- [Free gs-monoidal categories and free Markov categories](https://mlsanity.com/paper/2204.02284/) (2022)
- [The Logical Essentials of Bayesian Reasoning](https://mlsanity.com/paper/1804.01193/) (2018)
- [Interacting Hopf Algebras: the theory of linear systems](https://mlsanity.com/paper/1805.03032/) (2018)
- [Partial Markov Categories](https://mlsanity.com/paper/2502.03477/) (2025)
- [Custom Hypergraph Categories via Generalized Relations](https://mlsanity.com/paper/1703.01204/) (2017)

## Built on by (most cited first, 1 of 1)

- [Bayesian Networks, Markov Networks, Moralisation, Triangulation: a Categorical Perspective](https://mlsanity.com/paper/2512.09908/) (2025)

## Abstract (arXiv)

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 works in 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 are definable inductively on such syntax, and operate as a form of functor pre-composition. This approach introduces a modular, algebraic perspective in the theory of probabilistic graphical models.