Equations defining probability tree models

Abstract

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 contextspecific 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.

Nonlinear algebra and applications

“…Hence our model is a curve in the triangle ∆ 2 . The ideal generated by the quadratic polynomial above is the vanishing ideal of the model and it encodes the staged tree model; see [77].…”

“…A sufficient statistic of toric models is often achieved effortlessly from the monomial parametrization of the model in statistics; see for instance hierarchical models [68,121,163], graphical models [97], and balanced staged tree models [77]. In other cases, a linear change of coordinates is needed to reveal the toric structure; see for instance conditional independence models and models arising from Bayesian networks [95], and some group based phylogenetics models [226,58].…”

Gröbner bases for staged trees

“…For this we use the rings From the definition it follows that for every staged tree (T , θ ), the toric staged tree ideal is contained in the staged tree model ideal; i.e., ker(ϕ T ) ⊂ ker(φ T ). It is not true in general that these two ideals are equal [6]. However, Theorem 10 in [6] states that if a staged tree (T , θ ) is balanced, then ker(ϕ T ) = ker(φ T ).…”

“…It is not true in general that these two ideals are equal [6]. However, Theorem 10 in [6] states that if a staged tree (T , θ ) is balanced, then ker(ϕ T ) = ker(φ T ). Combining this result with Theorem 2.14 we can obtain Gröbner bases for staged tree model ideals whose staged tree is balanced and stratified.…”

“…The staged tree T is considered in [6,Section 6] as an example of the possible unfolding of events in a cell culture. A thorough discussion of this example and its difference with other graphical models is also contained in [6,Section 6]. Here we explain how to obtain a Gröbner basis for ker(ϕ T ) using Corollary 5.14.…”