2025/12/10 by Antonio Lorenzin, Lorenzin, Antonio, Fabio Zanasi +1
Computer Science · Neuroscience · #Artificial Intelligence (cs.AI) #Bayesian Modeling and Causal Inference #Category Theory (math.CT) #Constraint Satisfaction and Optimization #Embodied and Extended Cognition #FOS: Computer and information sciences #FOS: Mathematics #Logic in Computer Science (cs.LO)
paper · pdf · doi:10.48550/arxiv.2512.09908
openalex publication_date 2025/12/10 · openalex created_date 2025/12/12 · openalex updated_date 2026/07/28
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.