Identifiability of Gaussian structural equation models with equal error variances

By Jan Peters, Peter Buhlmann. 2012

398 citations recorded by the source corpus.

We consider structural equation models in which variables can be written as a function of their parents and noise terms, which are assumed to be jointly independent. Corresponding to each structural equation model is a directed acyclic graph describing the relationships between the variables. In Gaussian structural equation models with linear functions, the graph can be identified from the joint distribution only up to Markov equivalence classes, assuming faithfulness. In this work, we prove full identifiability in the case where all noise variables have the same variance: the directed acyclic graph can be recovered from the joint Gaussian distribution. Our result has direct implications for causal inference: if the data follow a Gaussian structural equation model with equal error variances, then, assuming that all variables are observed, the causal structure can be inferred from observational data only. We propose a statistical method and an algorithm based on our theoretical findings.

Subfields: Artificial Intelligence, Statistics and Probability, Causal Inference & Discovery

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Citations in the corpus

Gradient-Based Neural DAG Learning (2019, background): “There are many examples of graph identifiability results for continuous variables [11, 9, 14, 17] as well as for discrete variables [10].”
Algorithmic recourse under imperfect causal knowledge: a probabilistic approach (2020, methodology): “It is interesting to note, however, that the CVAE - SCM from (13) can be understood as a non-linear extension of the linear Gaussian model with equal error variances considered by [29], for which identiﬁability has been shown.”
Invariant Causal Prediction for Nonlinear Models (2017, background): “Such assumptions include non-Gaussianity (Shimizu et al., 2006) or equal variances (Peters and B¨uhlmann, 2014) of the errors or non-linearity of the structural equations in additive noise models (Hoyer et al., 2009; Peters et al., 2014).”
Causal inference by using invariant prediction: identification and confidence intervals (2015, background): “…framework of structural equation models, interesting work has been done for fully identiﬁable structures exploiting additional restrictions such as non-Gaussianity [Shimizu et al., 2006], nonlinearity [Hoyer et al., 2009, Peters et al., 2014] or equal error variances [Peters and B¨uhlmann, 2014].”
Distinguishing Cause from Effect Using Observational Data: Methods and Benchmarks (2014, background): “…et al., 2010; Mooij et al., 2010; Daniušis et al., 2010; Mooij et al., 2011; Shimizu et al., 2011; Janzing et al., 2012; Hyvärinen and Smith, 2013; Peters and Bühlmann, 2014; Kpotufe et al., 2014; Nowzohour and Bühlmann, 2015; Sgouritsa et al., 2015) that were claimed to be able to solve…”
CAM: Causal Additive Models, high-dimensional order search and penalized regression (2013, background): “Gaussian case, but this is not true anymore when placing additional restrictions which can be very useful [25, 26, 32].”
Causal Discovery with Reinforcement Learning (2019, methodology): “This setting is known to be identifiable according to Peters et al. (2014). We use a setup that is also considered by GraN-DAG (Lachapelle et al.”
Structural Intervention Distance (SID) for Evaluating Causal Graphs (2013, methodology): “The results also show how much can be gained when additional assumptions are appropriate; all methods exploit that the data come from a linear Gaussian SEM while only GDSEEV makes use of the additional constraint of equal error variances, which leads to identifiability of the DAG from the distributi”
Structural Intervention Distance for Evaluating Causal Graphs (2015, methodology): “This setting allows us to use the PC algorithm (Spirtes et al., 2000), conservative PC (Ramsey, Zhang, & Spirtes, 2006), greedy equivalent search (GES) (Chickering, 2002) and greedy DAG search based on the assumption of equal error variances (GDS EEV ) (Peters & B¨uhlmann, 2014).”
Score-based causal learning in additive noise models (2013, methodology): “Two other score-based methods have recently been proposed: for the parametric setting of Gaussian and linear models with same error variances [Peters and Bühlmann, 2012] and for linear models with non-Gaussian noise [Hyvärinen and Smith, 2013].”
DYNOTEARS: Structure Learning from Time-Series Data (2020, background): “Identiﬁabil-ity in this model is a well-known consequence of Marcinkiewicz’s theorem on the cumulants of the normal distribution (Kagan et al., 1973; Marcinkiewicz, 1939) • The errors z m,t are standard Gaussian, i.e. z m,t ∼ N (0 , I ). immediate Identiﬁability of this model is an immediate consequ”
Distinguishing Cause from Effect Using Observational Data: Methods and Benchmarks (2014, background): “…et al., 2010; Mooij et al., 2010; Daniušis et al., 2010; Mooij et al., 2011; Shimizu et al., 2011; Janzing et al., 2012; Hyvärinen and Smith, 2013; Peters and Bühlmann, 2014; Kpotufe et al., 2014; Nowzohour and Bühlmann, 2015; Sgouritsa et al., 2015) that were claimed to be able to solve…”
Structure Learning in Graphical Modeling (2016, background): “Identiﬁability of the DAG can also be achieved by various other restrictions on SEMs with additive noise, such as non-linear structural equations (Hoyer et al., 2009), additive models (B¨uhlmann et al., 2014), or equal error variances (Peters & B¨uhlmann, 2014).”
Learning Causal Structures Using Regression Invariance (2017, background): “There are studies which attempt to identify the exact causal structure by restricting the model class [32, 12, 24, 22].”
MAXIMUM LIKELIHOOD FOR SPARSE DIRECTED ACYCLIC GRAPHS (2013, background): “Then, the true DAG can be identified from the distribution [Peters and Bühlmann (2012)].”
Learning Directed Acyclic Graphs with Penalized Neighbourhood Regression (2015, background): “…(2013), who consider a similar (weighted) least squares estimator under the assumption that there exists an equivariance DAG, i.e. e e An equivariance DAG is known to be identiﬁable under certain assumptions (Peters and B¨uhlmann, 2013), whereas a minimum-trace DAG need not be identiﬁable.”
Masked Gradient-Based Causal Structure Learning (2019, methodology): “Other identifiable models include linear non-Gaussian model [29], linear Gaussian model with equal noise variances [23], post-nonlinear model [39], etc.”