Trek separation for Gaussian graphical models

Sullivant, Seth; Talaska, Kelli; Draisma, Jan

doi:10.1214/09-aos760

Cited by 57 publications

(78 citation statements)

References 12 publications

(16 reference statements)

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“…Pearl [1,20] already investigated constraints imposed by the special instrumental variable model. Furthermore, Darroch et al [21] and, recently, Sullivant et al [22] looked at linear Gaussian graphical models and determined constraints in terms of the entries on the covariance matrix describing the data (tetrad constraints). Further, methods of algebraic statistics were applied (e.g., [23]) to derive constraints that are induced by latent variable models directly on the level of probabilities.…”

Section: Discussionmentioning

confidence: 99%

Information-Theoretic Inference of Common Ancestors

Steudel

2015

Entropy

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A directed acyclic graph (DAG) partially represents the conditional independence structure among observations of a system if the local Markov condition holds, that is if every variable is independent of its non-descendants given its parents. In general, there is a whole class of DAGs that represents a given set of conditional independence relations. We are interested in properties of this class that can be derived from observations of a subsystem only. To this end, we prove an information-theoretic inequality that allows for the inference of common ancestors of observed parts in any DAG representing some unknown larger system. More explicitly, we show that a large amount of dependence in terms of mutual information among the observations implies the existence of a common ancestor that distributes this information. Within the causal interpretation of DAGs, our result can be seen as a quantitative extension of Reichenbach's principle of common cause to more than two variables. Our conclusions are valid also for non-probabilistic observations, such as binary strings, since we state the proof for an axiomatized notion of "mutual information" that includes the stochastic as well as the algorithmic version.

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Section: Discussionmentioning

confidence: 99%

Information-Theoretic Inference of Common Ancestors

Steudel

2015

Entropy

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“…Those yield pentad, and even higher order constraints. Sullivant, Talaska, and & Draisma (2010) give a detailed account of such constraints and show that both t-separation and d-separation can be derived from their general definitions.…”

Section: Effect Size and Tests For D-separation Constraintsmentioning

confidence: 99%

Local fit evaluation of structural equation models using graphical criteria.

Thoemmes¹,

Rosseel²,

Textor³

2018

Psychological Methods

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Evaluation of model fit is critically important for every structural equation model and sophisticated methods have been developed for this task. Among them are the χ 2 goodness-of-fit test, decomposition of the χ 2 , derived measures like the popular RMSEA or CFI, or inspection of residuals or modification indices. Many of these methods provide a global approach to model fit evaluation: A single index is computed that quantifies the fit of the entire SEM to the data. In contrast, graphical criteria like d-separation or trek-separation allow to derive implications that can be used for local fit evaluation, an approach that is hardly ever applied. We provide an overview of local fit evaluation from the viewpoint of SEM practitioners. In the presence of model misfit, local fit evaluation can potentially help in pinpointing where the problem with the model lies. For models that do fit the data, local tests can identify the parts of the model that are corroborated by the data. Local tests can also be conducted before a model is fitted at all, and they can be used even for models that are globally under-identified. We discuss appropriate statistical local tests, and provide applied examples. We also present novel software in R that automates this type of local fit evaluation.

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“…Our algorithm utilizes trek separation theorems that were originally proven by Sullivent [11], and then extended by Spirtes [9]. In order to understand why our algorithm works, it is first necessary to understand the trek separation theorems.…”

Section: Trek Separationmentioning

confidence: 99%

“…The definition of linear acyclicity (LA) below a choke set is complicated and is described in detail in [9]; for the purposes of this paper it suffices to note that, roughly, a directed graphical model is LA below sets (C A The following two theorems from [9] (extensions of theorems in [11]) relate the structure of the causal graph to the rank of the determinant of sub-matrices of the covariance matrix. …”

Section: Trek Separationmentioning

confidence: 99%

Causal Clustering for 2-Factor Measurement Models

Kummerfeld¹,

Ramsey²,

Yang³

et al. 2014

Machine Learning and Knowledge Discovery in Databases

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Many scientific research programs aim to learn the causal structure of real world phenomena. This learning problem is made more difficult when the target of study cannot be directly observed. One strategy commonly used by social scientists is to create measurable "indicator" variables that covary with the latent variables of interest. Before leveraging the indicator variables to learn about the latent variables, however, one needs a measurement model of the causal relations between the indicators and their corresponding latents. These measurement models are a special class of Bayesian networks. This paper addresses the problem of reliably inferring measurement models from measured indicators, without prior knowledge of the causal relations or the number of latent variables. We present a provably correct novel algorithm, FindOneFactorClusters (FOFC), for solving this inference problem. Compared to other state of the art algorithms, FOFC is faster, scales to larger sets of indicators, and is more reliable at small sample sizes. We also present the first correctness proofs for this problem that do not assume linearity or acyclicity among the latent variables.

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Trek separation for Gaussian graphical models

Cited by 57 publications

References 12 publications

Information-Theoretic Inference of Common Ancestors

Information-Theoretic Inference of Common Ancestors

Local fit evaluation of structural equation models using graphical criteria.

Causal Clustering for 2-Factor Measurement Models

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