Antonio Lorenzin

Tobias Fritz, Tomáš Gonda, Antonio Lorenzin et al.

Markov categories have recently turned out to be a powerful high-level framework for probability and statistics. They accommodate purely categorical definitions of notions like conditional probability and almost sure equality, as well as proofs of fundamental results such as the Hewitt--Savage 0/1 Law, the de Finetti Theorem and the Ergodic Decomposition Theorem. In this work, we develop additional relevant notions from probability theory in the setting of Markov categories. This comprises improved versions of previously introduced definitions of absolute continuity and supports, as well as a detailed study of idempotents and idempotent splitting in Markov categories. Our main result on idempotent splitting is that every idempotent measurable Markov kernel between standard Borel spaces splits across another standard Borel space, and we derive this as an instance of a general categorical criterion for idempotent splitting in Markov categories.

Antonio Lorenzin, Fabio Zanasi

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.