Telling cause from effect based on high-dimensional observations

By Dominik Janzing, Patrik O. Hoyer, Bernhard Schölkopf. 2009

81 citations recorded by the source corpus.

We describe a method for inferring linear causal relations among multi-dimensional variables. The idea is to use an asymmetry between the distributions of cause and effect that occurs if the covariance matrix of the cause and the structure matrix mapping the cause to the effect are independently chosen. The method applies to both stochastic and deterministic causal relations, provided that the dimensionality is sufficiently high (in some experiments, 5 was enough). It is applicable to Gaussian as well as non-Gaussian data.

Subfields: Artificial Intelligence, Causal Inference & Discovery, Statistical Learning Theory

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Citations in the corpus

Towards Causal Representation Learning (2021, background): “9Other dependence measures have been proposed for high-dimensional linear settings and time series [124, 226, 27, 122, 119, 125].”
A theory of independent mechanisms for extrapolation in generative models (2020, background): “(2015), the Trace Method (Janzing et al., 2010) that quantifies genericity with respect to the group of orthogonal transformations is the Fourier domain equivalent of SDR analysis.”
Causality for Machine Learning (2019, background): “It generalizes results of Shimizu et al. (2006) for linear functions, and it can be generalized to include nonlinear rescalings (Zhang and Hyvärinen, 2009), loops (Mooij et al., 2011), confounders (Janzing et al., 2009), and multi-variable settings (Peters et al., 2011).”
Group invariance principles for causal generative models (2017, methodology): “We show that this framework encompasses previous ICM approaches [Janzing et al., 2010; Shajarisales et al., 2015].”
Telling cause from effect in deterministic linear dynamical systems (2015, methodology): “The idea behind SIC, as well as several new approaches to causal inference (Daniusis et al., 2010; Janzing et al., 2010; Mooij et al., 2010; Zscheischler et al., 2011; Janzing et al., 2012; Sgouritsa et al., 2015), is to rely on the “philosophical” principle that the cause and”
Inferring deterministic causal relations (2010, methodology): “Hence, condition (7) is equivalent to ∆ = 0 and the restriction of the inference method in [Janzing et al., 2009] to deterministic linear models is a special case of our method.”
Distinguishing Cause from Effect Using Observational Data: Methods and Benchmarks (2014, background): “…2000; Kano and Shimizu, 2003; Shimizu et al., 2006; Sun et al., 2006, 2008; Hoyer et al., 2009; Mooij et al., 2009; Zhang and Hyvärinen, 2009; Janzing 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…”
Testing whether linear equations are causal: A free probability theory approach (2011, methodology): “(6) It is a finite sample analog to the quantity ∆X→Y in Janzing et al. (2010) and has been obtained by replacing the true covariance matrices with the empirical ones and by adding the factor n/r. Note that if P (X) has a density and k ≤ n it follows r = k − 1 almost surely, and r = n for k > n.”
Causal Discovery via Reproducing Kernel Hilbert Space Embeddings (2014, background): “To make the causal direction identifiable, it is assumed that A and x are generated independently and thus fulfill the trace condition (Janzing et al., 2010; Zscheischler et al., 2011), where x is the covariance matrix of x. Specifically, τm(A xA T ) ≈ τm(AAT )τn( x), where τm(·) = trace(·)/m.”
Causal Inference by Stochastic Complexity (2017, unknown): “…in Xn . en the distribution induced by the maximum likelihood parameters for Xn under the model classMm is given by P(Xn | θ̂ (Xn ,Mm )) = n∏ i=1 P(xi | θ̂ (Xn ,Mm )) = m∏ j=1 ( hj n )hj . e normalizing term R(Mm ,n) is given by R(Mm ,n) = ∑ Y n ∈Xn P(Yn | θ̂ (Yn ,Mm )) = ∑ h1+· · ·+hm=n n! h1!”
Probabilistic latent variable models for distinguishing between cause and effect (2010, background): “The challenge of inferring whether X causes Y (“X → Y ”) or vice versa (“Y → X”) from joint observations of the pair (X,Y ) has recently attracted increasing interest [1, 2, 3, 4, 5, 6, 7, 8].”
Estimating a Causal Order among Groups of Variables in Linear Models (2012, methodology): “[8] (i) conclude that ∆X→Y ≈ 0, (ii) show for the special case of B being an orthogonal matrix and the covariance matrix of e being λI, that ∆Y→X < 0, and (iii) show for the noise free case that ∆Y→X ≥ 0.”
Non-linear Causal Inference using Gaussianity Measures (2014, background): “Janzing et al. (2010) consider the problem of inferring linear causal relations among multi-dimensional variables.”
Information-geometric approach to inferring causal directions (2012, methodology): “[12] also considers the case Y = AX +E, where E is an independent noise term, but we restrict the attention to the deterministic one.”
Distinguishing Cause from Effect Using Observational Data: Methods and Benchmarks (2014, background): “…2000; Kano and Shimizu, 2003; Shimizu et al., 2006; Sun et al., 2006, 2008; Hoyer et al., 2009; Mooij et al., 2009; Zhang and Hyvärinen, 2009; Janzing 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…”
Origo: causal inference by compression (2016, methodology): “The linear trace method [21], [22] infers linear causal relations of the form Y = AX , where A is the structure matrix that maps the cause to the effect, using the linear trace condition which operates on A, and the covariance matrix of X , ΣX .”
Detecting Confounding in Multivariate Linear Models via Spectral Analysis (2017, methodology): “(16) almost coincide, while our method also accounts for higher order moments which the Trace Condition ignores.”