Gröbner bases for staged trees

Ananiadi, Lamprini; Duarte, Eliana

doi:10.2140/astat.2021.12.1

Cited by 15 publications

(22 citation statements)

References 16 publications

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“…Proof. We prove the first statement in (1). For this we show that Ψ TA (f ) = 0 for every f in the generating set of I M(TA) .…”

Section: Interventional Staged Tree Modelsmentioning

confidence: 80%

“…Using the main result in [1] we see that in fact the toric ideals in Theorem 5.4 and Corollary 5.6 have quadratic Gröbner bases with squarefree terms. We summarize this in the next theorem.…”

Section: Interventional Staged Tree Modelsmentioning

confidence: 86%

“…Let G = ([p], E) be a DAG. Then the following are equivalent: (1) The staged tree T G is balanced, (2) G is a perfect DAG, and (3)…”

Section: Corollary 38 Let G Be a Dag With Linear Extension πmentioning

confidence: 99%

“…For the part (2), let v, w be two nodes as in the statement of (2). Using (1) and the fact that v, w are in the same stage, we have p…”

Section: Interventional Staged Tree Modelsmentioning

confidence: 99%

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Algebraic geometry of discrete interventional models

Duarte¹,

Solus²

2020

Preprint

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We investigate the algebra and geometry of general interventions in discrete DAG models. To this end, we develop the formalism to study these models as subvarieties of multiprojective space and introduce a theory for modeling soft interventions in the more general family of staged tree models. We then consider the problem of finding their defining equations, and we derive a combinatorial criterion for identifying interventional staged tree models for which the defining ideal is toric. This criterion, when combined with a new characterization of decomposable DAG models in terms of their associated staged trees, specializes to a graphical criterion in the case of discrete interventional DAG models.

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“…Proof. We prove the first statement in (1). For this we show that Ψ TA (f ) = 0 for every f in the generating set of I M(TA) .…”

Section: Interventional Staged Tree Modelsmentioning

confidence: 80%

Section: Interventional Staged Tree Modelsmentioning

confidence: 86%

“…Let G = ([p], E) be a DAG. Then the following are equivalent: (1) The staged tree T G is balanced, (2) G is a perfect DAG, and (3)…”

Section: Corollary 38 Let G Be a Dag With Linear Extension πmentioning

confidence: 99%

“…For the part (2), let v, w be two nodes as in the statement of (2). Using (1) and the fact that v, w are in the same stage, we have p…”

Section: Interventional Staged Tree Modelsmentioning

confidence: 99%

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Algebraic geometry of discrete interventional models

Duarte¹,

Solus²

2020

Preprint

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“…. , X p ) is always balanced (see [1,Lemma 2.12]). The staged tree T G2 in Figure 1 is simple and therefore balanced.…”

Section: Balanced Modelsmentioning

confidence: 99%

A new characterization of discrete decomposable models

Duarte¹,

Solus²

2021

Preprint

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Decomposable graphical models, also known as perfect DAG models, play a fundamental role in standard approaches to probabilistic inference via graph representations in modern machine learning and statistics. However, such models are limited by the assumption that the data-generating distribution does not entail strictly context-specific conditional independence relations. The family of staged tree models generalizes DAG models so as to accommodate context-specific knowledge. We provide a new characterization of perfect discrete DAG models in terms of their staged tree representations. This characterization identifies the family of balanced staged trees as the natural generalization of discrete decomposable models to the context-specific setting.

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Families of Polytopes with Rational Linear Precision in Higher Dimensions

et al. 2022

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In this article, we introduce a new family of lattice polytopes with rational linear precision. For this purpose, we define a new class of discrete statistical models that we call multinomial staged tree models. We prove that these models have rational maximum likelihood estimators (MLE) and give a criterion for these models to be log-linear. Our main result is then obtained by applying Garcia-Puente and Sottile’s theorem that establishes a correspondence between polytopes with rational linear precision and log-linear models with rational MLE. Throughout this article, we also study the interplay between the primitive collections of the normal fan of a polytope with rational linear precision and the shape of the Horn matrix of its corresponding statistical model. Finally, we investigate lattice polytopes arising from toric multinomial staged tree models, in terms of the combinatorics of their tree representations.

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Gröbner bases for staged trees

Cited by 15 publications

References 16 publications

Algebraic geometry of discrete interventional models

Algebraic geometry of discrete interventional models

A new characterization of discrete decomposable models

Families of Polytopes with Rational Linear Precision in Higher Dimensions

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