Acyclic Linear SEMs Obey the Nested Markov Property

The conditional independence structure induced on the observed marginal distribution by a hidden variable directed acyclic graph (DAG) may be represented by a graphical model represented by mixed graphs called maximal ancestral graphs (MAGs). This model has a number of desirable properties, in particular the set of Gaussian distributions can be parameterized by viewing the graph as a path diagram. Models represented by MAGs have been used for causal discovery [22], and identification theory for causal effects [28]. In addition to ordinary conditional independence constraints, hidden variable DAGs also induce generalized independence constraints. These constraints form the nested Markov property [20]. We first show that acyclic linear SEMs obey this property. Further we show that a natural parameterization for all Gaussian distributions obeying the nested Markov property arises from a generalization of maximal ancestral graphs that we call maximal arid graphs (MArG). We show that every nested Markov model can be associated with a MArG; viewed as a path diagram this MArG parametrizes the Gaussian nested Markov model. This leads directly to methods for ML fitting and computing BIC scores for Gaussian nested models.

[1]  T. Richardson.  Markov Properties for Acyclic Directed Mixed Graphs , 2003 .

[2]  Thomas S. Richardson,et al.  Smooth, identifiable supermodels of discrete DAG models with latent variables , 2015, Bernoulli.

[3]  Judea Pearl,et al.  Graphical Condition for Identification in recursive SEM , 2006, UAI.

[4]  P. Spirtes,et al.  Using Path Diagrams as a Structural Equation Modeling Tool , 1998 .

[5]  James M. Robins,et al.  Nested Markov Properties for Acyclic Directed Mixed Graphs , 2012, UAI.

[6]  Jiji Zhang,et al.  Generalized Do-Calculus with Testable Causal Assumptions , 2007, AISTATS.

[7]  P. Spirtes,et al.  Ancestral graph Markov models , 2002 .

[8]  Michael Eichler,et al.  Computing Maximum Likelihood Estimates in Recursive Linear Models with Correlated Errors , 2009, J. Mach. Learn. Res..

[9]  M. Drton,et al.  Half-trek criterion for generic identifiability of linear structural equation models , 2011, 1107.5552.

[10]  Jin Tian,et al.  Testable Implications of Linear Structural Equation Models , 2014, AAAI.

[11]  Judea Pearl,et al.  A graphical criterion for the identification of causal effects in linear models , 2002, AAAI/IAAI.

[12]  Tom Burr,et al.  Causation, Prediction, and Search , 2003, Technometrics.

[13]  Marco Valtorta,et al.  Pearl's Calculus of Intervention Is Complete , 2006, UAI.

[14]  Robin J. Evans,et al.  Distributional Equivalence and Structure Learning for Bow-free Acyclic Path Diagrams , 2015, 1508.01717.

[15]  Judea Pearl,et al.  Identification of Joint Interventional Distributions in Recursive Semi-Markovian Causal Models , 2006, AAAI.

[16]  Mathias Drton,et al.  Comments on: Sequences of regressions and their independencies , 2012 .

[17]  Seth Sullivant,et al.  Identifying Causal Effects with Computer Algebra , 2010, UAI.

[18]  D. A. Kenny,et al.  Correlation and Causation , 1937, Wilmott.

[19]  Bryant Chen,et al.  Identification and Overidentification of Linear Structural Equation Models , 2016, NIPS.

[20]  Luca Weihs,et al.  Generic Identifiability of Linear Structural Equation Models by Ancestor Decomposition , 2015, 1504.02992.

[21]  Jin Tian,et al.  Parameter Identification in a Class of Linear Structural Equation Models , 2009, IJCAI.

[22]  Joris M. Mooij,et al.  Algebraic Equivalence Class Selection for Linear Structural Equation Models , 2018, UAI.

[23]  T. Richardson,et al.  Markovian acyclic directed mixed graphs for discrete data , 2013, 1301.6624.

[24]  D. A. Kenny,et al.  Correlation and Causation. , 1982 .

[25]  Joris M. Mooij,et al.  Algebraic Equivalence of Linear Structural Equation Models , 2018, ArXiv.

[26]  T. Haavelmo.  The Statistical Implications of a System of Simultaneous Equations , 1943 .

[27]  J. Koster.  On the Validity of the Markov Interpretation of Path Diagrams of Gaussian Structural Equations Systems with Correlated Errors , 1999 .

[28]  M. Drton,et al.  Global identifiability of linear structural equation models , 2010, 1003.1146.

[29]  Judea Pearl,et al.  Equivalence and Synthesis of Causal Models , 1990, UAI.

[30]  J. Pearl.  Causality: Models, Reasoning and Inference , 2000 .