Positivity for Gaussian graphical models

Draisma, Jan; Sullivant, Seth; Talaska, Kelli

doi:10.1016/j.aam.2013.03.001

Cited by 12 publications

(25 citation statements)

References 9 publications

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“…This proof follows, essentially exactly, as the first part of the proof of Prop. 2.5 in Draisma et al (2013). Now we show that the above subdivision trick produces a graph G * flow for which the max-flow between vertex sets is the same as for G * flow .…”

Section: Resultsmentioning

confidence: 73%

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Determinantal Generalizations of Instrumental Variables

Weihs

Robinson

Dufresne

et al. 2017

Journal of Causal Inference

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Abstract. Linear structural equation models relate the components of a random vector using linear interdependencies and Gaussian noise. Each such model can be naturally associated with a mixed graph whose vertices correspond to the components of the random vector. The graph contains directed edges that represent the linear relationships between components, and bidirected edges that encode unobserved confounding. We study the problem of generic identifiability, that is, whether a generic choice of linear and confounding effects can be uniquely recovered from the joint covariance matrix of the observed random vector. An existing combinatorial criterion for establishing generic identifiability is the half-trek criterion (HTC), which uses the existence of trek systems in the mixed graph to iteratively discover generically invertible linear equation systems in polynomial time. By focusing on edges one at a time, we establish new sufficient and necessary conditions for generic identifiability of edge effects extending those of the HTC. In particular, we show how edge coefficients can be recovered as quotients of subdeterminants of the covariance matrix, which constitutes a determinantal generalization of formulas obtained when using instrumental variables for identification.

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Section: Resultsmentioning

confidence: 73%

“…Theorem 3.2(Sullivant et al (2010),Draisma et al (2013)). The submatrix Σ S,T has generic rank ≤ r if and only if there exist sets L, R…”

mentioning

confidence: 97%

Determinantal Generalizations of Instrumental Variables

Weihs

Robinson

Dufresne

et al. 2017

Journal of Causal Inference

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“…The following proposition captures exactly which terms cancel. For more details on the arguments, we refer to [STD10,DST13]. Moreover, for any two self-avoiding cup systems U and U ′ with w(U) = w(U ′ ) we have sgn (U) = sgn (U ′ ).…”

Section: Subdeterminants Of Concentration Matricesmentioning

confidence: 99%

Automorphism groups of Gaussian Bayesian networks

Draisma¹,

Zwiernik²

2017

Bernoulli

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In this paper we extend earlier work on groups acting on Gaussian graphical models to Gaussian Bayesian networks and more general Gaussian models defined by chain graphs. We discuss the maximal group which leaves a given model invariant and provide basic statistical applications of this result. This includes equivariant estimation, maximal invariants and robustness. The computation of the group requires finding the essential graph. However, by applying Stúdeny's theory of imsets we show that computations for DAGs can be performed efficiently without building the essential graph. In our proof we derive simple necessary and sufficient conditions on vanishing sub-minors of the concentration matrix in the model.

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“…In [STD10] the proposition is used to give a combinatorial criterion, generalising d-separation, for the determinant to be identically zero on R D × R V >0 . Furthermore, in [DST13] it is shown that the sum above is cancellation-free: if two trek systems I → J have the same weight, then they have the same sign. Moreover, it is shown there that the coefficient of each monomial is plus or minus a power of 2.…”

Section: Introductionmentioning

confidence: 99%

Partial correlation hypersurfaces in Gaussian graphical models

Draisma¹

2019

Algebraic Combinatorics

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We derive a combinatorial sufficient condition for a partial correlation hypersurface in the parameter space of a directed Gaussian graphical model to be nonsingular, and speculate on whether this condition can be used in algorithms for learning the graph. Since the condition is fulfilled in the case of a complete DAG on any number of vertices, the result implies an affirmative answer to a question raised by Lin-Uhler-Sturmfels-Bühlmann.

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Positivity for Gaussian graphical models

Cited by 12 publications

References 9 publications

Determinantal Generalizations of Instrumental Variables

Determinantal Generalizations of Instrumental Variables

Automorphism groups of Gaussian Bayesian networks

Partial correlation hypersurfaces in Gaussian graphical models

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