Equations defining probability tree models
Christiane Görgen and Eliana Maria Duarte Gelvez
Contact the author: Please use for correspondence this email.
Submission date: 23. Feb. 2018
MSC-Numbers: 62-0, 13P25
Keywords and phrases: algebraic statistics, graphical models, Staged TRees
Link to arXiv:See the arXiv entry of this preprint.
Coloured probability tree models are statistical models coding conditional independence between events depicted in a tree graph. They are more general than the very important class of context- specific Bayesian networks. In this paper, we study the algebraic properties of their ideal of model invariants. The generators of this ideal can be easily read from the tree graph and have a straightforward interpretation in terms of the underlying model: they are differences of odds ratios coming from conditional probabilities. One of the key findings in this analysis is that the tree is a convenient tool for understanding the exact algebraic way in which the sum-to-1 conditions on the parameter space translate into the sum-to-one conditions on the joint probabilities of the statistical model. This enables us to identify necessary and sufficient graphical conditions for a staged tree model to be a toric variety intersected with a probability simplex.