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

Antonio Lorenzin; Fabio Zanasi

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arXiv · 2512.09908

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

Antonio Lorenzin·Fabio Zanasi

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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.

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Antonio Lorenzin, Fabio Zanasi. 2025-12-10. Bayesian Networks, Markov Networks, Moralisation, Triangulation: a Categorical Perspective. https://arxiv.org/abs/2512.09908

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