An Algebraic Approach to Moralisation and Triangulation of Probabilistic Graphical Models
This provides a theoretical foundation for researchers in probabilistic graphical models, but it is incremental as it formalizes existing transformations without new empirical results.
The paper tackled the problem of transforming between Bayesian and Markov networks by developing a categorical framework where moralisation and triangulation are modeled as functors, introducing a modular, algebraic perspective in probabilistic graphical models.
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.