Equations defining probability tree models. (July 2020) - Duarte, Eliana; Görgen, Christiane
Abstract: Staged trees or coloured probability tree models are statistical models coding conditional independence between events depicted in a tree graph. They include the very important class of Bayesian networks as a special case and provide a straightforward graphical tool for handling additional context-specific relationships. In this paper, we study the algebraic properties of their ideal of model invariants. We hereby find that the tree also provides a straightforward combinatorial tool to generalise the existing geometric characterisation of decomposable graphical models and Bayesian networks. In particular, from a staged tree we can directly understand the interplay between local and global sum-to-one conditions, read the generators of that ideal, and determine conditions under which the model is a toric variety intersected with the probability simplex.