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

TL;DR

This paper introduces a categorical framework to model transformations between Bayesian and Markov networks, emphasizing the syntactic and semantic aspects of moralisation and triangulation.

Contribution

It formalizes moralisation and triangulation as functors within a categorical setting, providing a new perspective on probabilistic graphical model transformations.

Findings

Moralisation is fully syntactic, while triangulation depends on semantics.

A functorial interpretation of the variable elimination algorithm is proposed.

The framework clarifies the distinction between syntactic and semantic modifications.

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.