Identifiability of Causal Graphs using Functional Models

By Jan Peters, Joris M. Mooij, Dominik Janzing, Bernhard Schölkopf. 2011

174 citations recorded by the source corpus.

This work addresses the following question: Under what assumptions on the data generating process can one infer the causal graph from the joint distribution? The approach taken by conditional independence-based causal discovery methods is based on two assumptions: the Markov condition and faithfulness. It has been shown that under these assumptions the causal graph can be identified up to Markov equivalence (some arrows remain undirected) using methods like the PC algorithm. In this work we propose an alternative by defining Identifiable Functional Model Classes (IFMOCs). As our main theorem we prove that if the data generating process belongs to an IFMOC, one can identify the complete causal graph. To the best of our knowledge this is the first identifiability result of this kind that is not limited to linear functional relationships. We discuss how the IFMOC assumption and the Markov and faithfulness assumptions relate to each other and explain why we believe that the IFMOC assumption can be tested more easily on given data. We further provide a practical algorithm that recovers the causal graph from finitely many data; experiments on simulated data support the theoretical findings.

Subfields: Artificial Intelligence, Causal Inference & Discovery

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Citations in the corpus

Toward Causal Representation Learning (2021, background): “[229] for linear functions, and it can be generalized to include nonlinear rescalings [279], loops [174], confounders [123], and multivariable settings [189].”
Towards Causal Representation Learning (2021, background): “[237, 188, 229, 113, 174, 187, 139, 17, 246, 277, 175, 123, 186, 176, 36, 82, 161], learning causal relations frequently requires collecting data from multiple environments, or the ability to perform interventions [251].”
Causality for Machine Learning (2019, background): “(Image courtesy of Markus Elsholz, reprinted from Peters et al. (2017).)”
Learning Neural Causal Models from Unknown Interventions (2019, methodology): “Asymmetry-based methods [8, 32, 33, 34, 35, 36] assume asymmetry between cause and effect in the data and try to use this information to estimate the causal structure.”
Consistency of Causal Inference under the Additive Noise Model (2013, methodology): “Various works (Shimizu et al., 2006; Hoyer et al., 2009; Peters et al., 2011a) have successfully validated procedures based on the CAM (outlined in Section 1.1 below) on a mix of artificial and real-world datasets where the causal structure to be inferred is clear.”
Identifiability of Gaussian structural equation models with equal error variances (2012, methodology): “…G0 [Shimizu et al., 2006]; (ii) if one restricts the functions to be additive in the noise component and excludes the linear Gaussian case, as well as a few other pathological function-noise combinations, one can show that G0 is identifiable from L(X) [Hoyer et al., 2009, Peters et al., 2011].”
Cyclic Causal Discovery from Continuous Equilibrium Data (2013, background): “Recently, several methods have been proposed for causal discovery from continuous-valued observational data by exploiting independence of the estimated noise with the input (Shimizu et al., 2006; Hoyer et al., 2009; Zhang and Hyv¨arinen, 2009; Peters et al., 2011).”
Score-based causal learning in additive noise models (2013, background): “We also look at two simulation settings suggested in [10], where the graph consists of four nodes and the edge functions are nonlinear but parametrized by four and five parameters, respectively.”
Identification of Time-Dependent Causal Model: A Gaussian Process Treatment (2015, background): “This directly implies that based on the formulated timedependent nonlinear causal model (3), the causal structure is identifiable under mild assumptions (for details of the assumptions, see [Hoyer et al., 2009; Zhang and Hyvärinen, 2009b; Peters et al., 2011]).”
Causal Inference on Time Series using Structural Equation Models (2012, background): “This idea has led to the extensions of nonlinear additive functions f ( x, n ) = g ( x ) + n [Hoyer et al., 2009], post-nonlinear additive functions f ( x, n ) = h g ( x ) + n [Zhang and Hyvarinen, 2009] and discrete functions [Peters et al., 2011a].”
Causal Discovery and Inference: Concepts and Recent Methodological Advances (2016, background): “Below we focus on the two-variable case, and the results can be readily extended to the case with an arbitrary number of variables, as shown in Peters et al. (2011).”
Causal Inference on Time Series using Restricted Structural Equation Models (2013, background): “For large sample sizes, even smallest differences between the true data generating process and the model may lead to rejected independence tests [discussed by Peters et al., 2011a] .”
UvA-DARE ( Digital Academic Repository ) Causal Discovery with Continuous Additive Noise Models (2014, unknown): “This paper builds on the conference papers of Hoyer et al. (2009), Peters et al. (2011b) and Mooij et al. (2009) 1 but extends the material in several aspects.”
Robust Learning via Cause-Effect Models (2011, background): “We assume that N C and N E are independent, in which case we may restrict our attention to the present graph (causal suﬃciency).”
Causal network inference using biochemical kinetics (2014, unknown): “Non-linear kinetic equations, in contrast, are able to confer asymmetries between nodes and may be sufficient to enable orientation of edges (Peters et al. , 2011), although we note that causal inference using non-linear models still requires a number of strong assumptions (Pearl, 2009).”
Jointly interventional and observational data: estimation of interventional Markov equivalence classes of directed acyclic graphs (2013, methodology): “Under additional assumptions, identiﬁability of the whole DAG is guaranteed as with linear structural equation models with non-Gaussian errors (Shimizu et al., 2006) or additive noise models (Hoyer et al., 2009), see also Peters et al. (2011).”
Causal inference in partially linear structural equation models (2016, unknown): “Based on bivariate identifiability, it has been shown that one can conclude multivariate identifiability under an additional assumption referred to as IFMOC assumption [17].”