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Halftrek criterion for generic identifiability of linear structural equation models
 ANNALS OF STATISTICS, TO APPEAR
, 2011
"... A linear structural equation model relates random variables of interest and corresponding Gaussian noise terms via a linear equation system. Each such model can be represented by a mixed graph in which directed edges encode the linear equations, and bidirected edges indicate possible correlations ..."
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A linear structural equation model relates random variables of interest and corresponding Gaussian noise terms via a linear equation system. Each such model can be represented by a mixed graph in which directed edges encode the linear equations, and bidirected edges indicate possible correlations among noise terms. We study parameter identifiability in these models, that is, we ask for conditions that ensure that the edge coefficients and correlations appearing in a linear structural equation model can be uniquely recovered from the covariance matrix of the associated normal distribution. We treat the case of generic identifiability, where unique recovery is possible for almost every choice of parameters. We give a new graphical criterion that is sufficient for generic identifiability. It improves criteria from prior work and does not require the directed part of the graph to be acyclic. We also develop a related necessary condition and examine the “gap ” between sufficient and necessary conditions through simulations as well as exhaustive algebraic computations for graphs with up to five nodes.
Counterfactual analyses with graphical models based on local independence
, 2012
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Linear Social Interactions Models
, 2013
"... three referees for very helpful comments and suggestions. Wallice Ao, Joel Han, Hon Ho Kwok, Ariel Roginsky, Kegon Tan and Xiangrong Yu have provided superb research assistance. We are grateful ..."
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three referees for very helpful comments and suggestions. Wallice Ao, Joel Han, Hon Ho Kwok, Ariel Roginsky, Kegon Tan and Xiangrong Yu have provided superb research assistance. We are grateful
Identification of discrete concentration graph models with one hidden binary variable
"... Conditions are presented for different types of identifiability of discrete variable models generated over an undirected graph in which one node represents a binary hidden variable. These models can be seen as extensions of the latent class model to allow for conditional associations between the obs ..."
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Conditions are presented for different types of identifiability of discrete variable models generated over an undirected graph in which one node represents a binary hidden variable. These models can be seen as extensions of the latent class model to allow for conditional associations between the observable random variables. Since local identification corresponds to full rank of the parametrization map, we establish a necessary and sufficient condition for the rank to be full everywhere in the parameter space. The condition is based on the topology of the undirected graph associated to the model. For nonfull rank models, the obtained characterization allows us to find the subset of the parameter space where the identifiability breaks down.
Invited Commentary Invited Commentary: Structural Equation Models and Epidemiologic Analysis
, 2012
"... In this commentary, structural equation models (SEMs) are discussed as a tool for epidemiologic analysis. Such models are related to and compared with other analytic approaches often used in epidemiology, including regression analysis, causal diagrams, causal mediation analysis, and marginal structu ..."
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In this commentary, structural equation models (SEMs) are discussed as a tool for epidemiologic analysis. Such models are related to and compared with other analytic approaches often used in epidemiology, including regression analysis, causal diagrams, causal mediation analysis, and marginal structural models. Several of these other approaches in fact developed out of the SEM literature. However, SEMs themselves tend to make much stronger assumptions than these other techniques. SEMs estimate more types of effects than do these other techniques, but this comes at the price of additional assumptions. Many of these assumptions have often been ignored and not carefully evaluated when SEMs have been used in practice. In light of the strong assumptions employed by SEMs, the author argues that they should be used principally for the purposes of exploratory analysis and hypothesis generation when a broad range of effects are potentially of interest. causal inference; causality; causal modeling; confounding factors (epidemiology); epidemiologic methods; regression analysis; structural equation model Abbreviations: MSM, marginal structural model; SEM structural equation model. Every so often a paper is published in the epidemiologic literature that employs structural equation models (SEMs). Sometimes the authors of these papers advocate that such
POSITIVITY FOR GAUSSIAN GRAPHICAL MODELS
, 2012
"... Gaussian graphical models are parametric statistical models for jointly normal random variables whose dependence structure is determined by a graph. In previous work, we introduced trek separation, which gives a necessary and sufficient condition in terms of the graph for when a subdeterminant is ..."
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Gaussian graphical models are parametric statistical models for jointly normal random variables whose dependence structure is determined by a graph. In previous work, we introduced trek separation, which gives a necessary and sufficient condition in terms of the graph for when a subdeterminant is zero for all covariance matrices that belong to the Gaussian graphical model. Here we extend this result to give explicit cancellationfree formulas for the expansions of nonzero subdeterminants.
In Revision
, 2012
"... Economic Thinking, all of which is greatly appreciated. Hon Ho Kwok and Xiangrong Yu have provided superb research assistance. We are grateful for comments from Youcef Msaid, Alex ReesJones, Dean Robinson, Michael Strain and Nichole Szembrot and to Charles Manski and Hashem Pesaran for discussions ..."
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Economic Thinking, all of which is greatly appreciated. Hon Ho Kwok and Xiangrong Yu have provided superb research assistance. We are grateful for comments from Youcef Msaid, Alex ReesJones, Dean Robinson, Michael Strain and Nichole Szembrot and to Charles Manski and Hashem Pesaran for discussions. This paper was written in honor of James J. Heckman, whose influence will be evident throughout.
TypeII Errors of Independence Tests Can Lead to Arbitrarily Large Errors in Estimated Causal Effects: An Illustrative Example
"... Estimating the strength of causal effects from observational data is a common problem in scientific research. A popular approach is based on exploiting observed conditional independences between variables. It is wellknown that this approach relies on the assumption of faithfulness. In our opinion ..."
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Estimating the strength of causal effects from observational data is a common problem in scientific research. A popular approach is based on exploiting observed conditional independences between variables. It is wellknown that this approach relies on the assumption of faithfulness. In our opinion, a more important practical limitation of this approach is that it relies on the ability to distinguish independences from (arbitrarily weak) dependences. We present a simple analysis, based on purely algebraic and geometrical arguments, of how the estimation of the causal effect strength, based on conditional independence tests and background knowledge, can have an arbitrarily large error due to the uncontrollable type II error of a single conditional independence test. The scenario we are studying here is related to the LCD algorithm by Cooper [1] and to the instrumental variable setting that is popular in epidemiology and econometry. It is one of the simplest settings in which causal discovery and prediction methods based on conditional independences arrive at nontrivial conclusions, yet for which the lack of uniform consistency can result in arbitrarily large prediction errors.